Exhaustion by compact setsFrom Wikipedia, the free encyclopedia

Contents

1 a-paracompact space 11.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Binary relation 22.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1.1 Is a relation more than its graph? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Special types of binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2.1 Difunctional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Relations over a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Operations on binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.4.1 Complement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4.2 Restriction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4.3 Algebras, categories, and rewriting systems . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.5 Sets versus classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.6 The number of binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.7 Examples of common binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.11 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Closed set 123.1 Equivalent definitions of a closed set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Properties of closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3 Examples of closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.4 More about closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4 Closure (topology) 144.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.1.1 Point of closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

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4.1.2 Limit point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.1.3 Closure of a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.3 Closure operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.4 Facts about closures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.5 Categorical interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5 Compact operator 185.1 Equivalent formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.2 Important properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.3 Origins in integral equation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.4 Compact operator on Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.5 Completely continuous operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

6 Compact space 226.1 Historical development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.2 Basic examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.3 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

6.3.1 Open cover definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.3.2 Equivalent definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.3.3 Compactness of subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

6.4 Properties of compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.4.1 Functions and compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.4.2 Compact spaces and set operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.4.3 Ordered compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

6.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.5.1 Algebraic examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

6.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

7 Compactly embedded 317.1 Definition (topological spaces) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

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7.2 Definition (normed spaces) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

8 Cover (topology) 328.1 Cover in topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328.2 Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328.3 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.4 Covering dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

9 Euclidean space 359.1 Intuitive overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359.2 Euclidean structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

9.2.1 Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379.2.2 Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389.2.3 Rotations and reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389.2.4 Euclidean group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

9.3 Non-Cartesian coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409.4 Geometric shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

9.4.1 Lines, planes, and other subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419.4.2 Line segments and triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.4.3 Polytopes and root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.4.4 Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.4.5 Balls, spheres, and hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

9.5 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449.7 Alternatives and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

9.7.1 Curved spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449.7.2 Indefinite quadratic form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449.7.3 Other number fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459.7.4 Infinite dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

9.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459.9 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459.11 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

10 Exhaustion by compact sets 4610.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4610.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

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10.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

11 Feebly compact space 47

12 Functional analysis 4812.1 Normed vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

12.1.1 Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4912.1.2 Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

12.2 Major and foundational results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4912.2.1 Uniform boundedness principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5012.2.2 Spectral theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5012.2.3 Hahn-Banach theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5012.2.4 Open mapping theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5112.2.5 Closed graph theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5112.2.6 Other topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

12.3 Foundations of mathematics considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5112.4 Points of view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5112.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5212.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5212.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5212.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

13 H-closed space 5413.1 Examples and equivalent formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5413.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5413.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

14 Hasse diagram 5514.1 A “good” Hasse diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5614.2 Upward planarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5614.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5614.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5714.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

15 Hausdorff space 5915.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5915.2 Equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6015.3 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6015.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6015.5 Preregularity versus regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6115.6 Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6115.7 Algebra of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6215.8 Academic humour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

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15.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6215.10Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6215.11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

16 Hemicompact space 6316.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6316.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6316.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6316.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

17 Interior (topology) 6517.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

17.1.1 Interior point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6617.1.2 Interior of a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

17.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6617.3 Interior operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6717.4 Exterior of a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6717.5 Interior-disjoint shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6817.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6817.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6817.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

18 k-cell (mathematics) 7018.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7018.2 Intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7018.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

19 Lebesgue covering dimension 7219.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7219.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7219.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7219.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7319.5 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

19.5.1 Historical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7319.5.2 Modern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

19.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7319.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

20 Limit point compact 7420.1 Properties and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7420.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7420.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7520.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

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21 Lindelöf space 7621.1 Properties of Lindelöf spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7621.2 Properties of strongly Lindelöf spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7621.3 Product of Lindelöf spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7621.4 Generalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7721.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7721.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7721.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

22 Locally compact space 7822.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7822.2 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

22.2.1 Compact Hausdorff spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7922.2.2 Locally compact Hausdorff spaces that are not compact . . . . . . . . . . . . . . . . . . . 7922.2.3 Hausdorff spaces that are not locally compact . . . . . . . . . . . . . . . . . . . . . . . . 7922.2.4 Non-Hausdorff examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

22.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8022.3.1 The point at infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8022.3.2 Locally compact groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

22.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8122.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

23 Locally finite 82

24 Locally finite collection 8324.1 Examples and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

24.1.1 Compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8324.1.2 Second countable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

24.2 Closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8424.3 Countably locally finite collections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8424.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

25 Locally finite space 8525.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

26 Manifold 8626.1 Motivational examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

26.1.1 Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8726.1.2 Other curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9026.1.3 Enriched circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

26.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9126.2.1 Early development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9126.2.2 Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

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26.2.3 Poincaré's definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9226.2.4 Topology of manifolds: highlights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

26.3 Mathematical definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9326.3.1 Broad definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

26.4 Charts, atlases, and transition maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9426.4.1 Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9426.4.2 Atlases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9426.4.3 Transition maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9426.4.4 Additional structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

26.5 Manifold with boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9526.5.1 Boundary and interior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

26.6 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9526.6.1 Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9526.6.2 Patchwork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9626.6.3 Identifying points of a manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9726.6.4 Gluing along boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9726.6.5 Cartesian products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

26.7 Manifolds with additional structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9726.7.1 Topological manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9726.7.2 Differentiable manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9826.7.3 Riemannian manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9826.7.4 Finsler manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9926.7.5 Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9926.7.6 Other types of manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

26.8 Classification and invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9926.9 Examples of surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

26.9.1 Orientability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10026.9.2 Genus and the Euler characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

26.10Maps of manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10126.10.1 Scalar-valued functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

26.11Generalizations of manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10226.12See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

26.12.1 By dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10326.13Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10326.14References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10426.15External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

27 Mathematical analysis 11127.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11227.2 Important concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

27.2.1 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11327.2.2 Sequences and limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

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27.3 Main branches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11427.3.1 Real analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11427.3.2 Complex analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11427.3.3 Functional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11427.3.4 Differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11427.3.5 Measure theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11527.3.6 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

27.4 Other topics in mathematical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11527.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

27.5.1 Physical sciences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11627.5.2 Signal processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11627.5.3 Other areas of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

27.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11627.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11727.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11827.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

28 Mesocompact space 11928.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11928.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

29 Metacompact space 12029.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12029.2 Covering dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12029.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12029.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

30 Metric space 12230.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12230.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12230.3 Examples of metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12330.4 Open and closed sets, topology and convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 12430.5 Types of metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

30.5.1 Complete spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12430.5.2 Bounded and totally bounded spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12530.5.3 Compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12630.5.4 Locally compact and proper spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12630.5.5 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12630.5.6 Separable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

30.6 Types of maps between metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12630.6.1 Continuous maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12730.6.2 Uniformly continuous maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

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30.6.3 Lipschitz-continuous maps and contractions . . . . . . . . . . . . . . . . . . . . . . . . . 12730.6.4 Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12830.6.5 Quasi-isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

30.7 Notions of metric space equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12830.8 Topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12830.9 Distance between points and sets; Hausdorff distance and Gromov metric . . . . . . . . . . . . . . 12930.10Product metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

30.10.1 Continuity of distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12930.11Quotient metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13030.12Generalizations of metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

30.12.1 Metric spaces as enriched categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13030.13See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13130.14Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13130.15References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13230.16External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

31 Metrization theorem 13331.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13331.2 Metrization theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13331.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13431.4 Examples of non-metrizable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13431.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13431.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

32 Normal space 13532.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13532.2 Examples of normal spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13632.3 Examples of non-normal spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13632.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13732.5 Relationships to other separation axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13732.6 Citations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13732.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

33 Open set 13833.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13933.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

33.2.1 Euclidean space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14033.2.2 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14033.2.3 Topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

33.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14033.4 Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14033.5 Notes and cautions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

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33.5.1 “Open” is defined relative to a particular topology . . . . . . . . . . . . . . . . . . . . . . 14133.5.2 Open and closed are not mutually exclusive . . . . . . . . . . . . . . . . . . . . . . . . . 141

33.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14133.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14133.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

34 Order theory 14334.1 Background and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14334.2 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

34.2.1 Partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14434.2.2 Visualizing a poset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14434.2.3 Special elements within an order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14434.2.4 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14634.2.5 Constructing new orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

34.3 Functions between orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14634.4 Special types of orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14734.5 Subsets of ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14834.6 Related mathematical areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

34.6.1 Universal algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14834.6.2 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14834.6.3 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

34.7 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14934.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14934.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14934.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14934.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

35 Orthocompact space 15135.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

36 Paracompact space 15236.1 Paracompactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15236.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15236.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15336.4 Paracompact Hausdorff Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

36.4.1 Partitions of unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15436.5 Relationship with compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

36.5.1 Comparison of properties with compactness . . . . . . . . . . . . . . . . . . . . . . . . . 15536.6 Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

36.6.1 Definition of relevant terms for the variations . . . . . . . . . . . . . . . . . . . . . . . . . 15636.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15636.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

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36.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15736.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

37 Partially ordered set 15837.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15937.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15937.3 Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15937.4 Orders on the Cartesian product of partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . 16037.5 Sums of partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16037.6 Strict and non-strict partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16137.7 Inverse and order dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16137.8 Mappings between partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16137.9 Number of partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16237.10Linear extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16237.11In category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16337.12Partial orders in topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16337.13Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16337.14See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16337.15Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16437.16References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16437.17External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

38 Partition of unity 16538.1 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16538.2 Variant definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16638.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16638.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16638.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16638.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

39 Product topology 16739.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16739.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16739.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16839.4 Relation to other topological notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16939.5 Axiom of choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16939.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16939.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16939.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17039.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

40 Pseudocompact space 17140.1 Properties related to pseudocompactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

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40.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17140.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

41 Realcompact space 17341.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17341.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17341.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

42 Regular space 17542.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17542.2 Relationships to other separation axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17642.3 Examples and nonexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17642.4 Elementary properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17742.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

43 Relatively compact subspace 17843.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17843.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

44 Second-countable space 17944.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

44.1.1 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17944.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18044.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

45 Sequence 18145.1 Examples and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

45.1.1 Important examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18245.1.2 Indexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18345.1.3 Specifying a sequence by recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

45.2 Formal definition and basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18445.2.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18445.2.2 Finite and infinite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18545.2.3 Increasing and decreasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18545.2.4 Bounded . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18545.2.5 Other types of sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

45.3 Limits and convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18645.3.1 Definition of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18745.3.2 Applications and important results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18745.3.3 Cauchy sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

45.4 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18845.5 Use in other fields of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

45.5.1 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

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45.5.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18945.5.3 Linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19045.5.4 Abstract algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19045.5.5 Set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19145.5.6 Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19145.5.7 Streams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

45.6 Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19145.7 Related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19245.8 Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19245.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19245.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19245.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

46 Sequentially compact space 19446.1 Examples and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19446.2 Related notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19446.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19446.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19446.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

47 Set (mathematics) 19647.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19747.2 Describing sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19747.3 Membership . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

47.3.1 Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19947.3.2 Power sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

47.4 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20047.5 Special sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20047.6 Basic operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

47.6.1 Unions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20147.6.2 Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20247.6.3 Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20247.6.4 Cartesian product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

47.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20547.8 Axiomatic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20547.9 Principle of inclusion and exclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20647.10De Morgan’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20647.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20747.12Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20747.13References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20747.14External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

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48 Strictly singular operator 20848.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

49 Subset 20949.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21049.2 ⊂ and ⊃ symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21049.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21049.4 Other properties of inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21149.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21149.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21149.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

50 Subspace topology 21350.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21350.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21350.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21450.4 Preservation of topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21550.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21550.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

51 Supercompact space 21651.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21651.2 Some Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21651.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

52 Topological space 21852.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

52.1.1 Neighbourhoods definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21852.1.2 Open sets definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21952.1.3 Closed sets definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22052.1.4 Other definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

52.2 Comparison of topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22052.3 Continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22052.4 Examples of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22152.5 Topological constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22252.6 Classification of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22252.7 Topological spaces with algebraic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22252.8 Topological spaces with order structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22252.9 Specializations and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22252.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22352.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22352.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22352.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

CONTENTS xv

53 Topology 22553.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22653.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22753.3 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

53.3.1 Topologies on Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22953.3.2 Continuous functions and homeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . 23053.3.3 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

53.4 Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23053.4.1 General topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23053.4.2 Algebraic topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23153.4.3 Differential topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23153.4.4 Geometric topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23153.4.5 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

53.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23253.5.1 Biology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23253.5.2 Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23253.5.3 Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23253.5.4 Robotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

53.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23253.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23353.8 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23453.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

54 Total order 23554.1 Strict total order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23554.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23654.3 Further concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

54.3.1 Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23654.3.2 Lattice theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23654.3.3 Finite total orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23754.3.4 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23754.3.5 Order topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23754.3.6 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23754.3.7 Sums of orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

54.4 Orders on the Cartesian product of totally ordered sets . . . . . . . . . . . . . . . . . . . . . . . . 23854.5 Related structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23854.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23854.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23854.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

55 Totally bounded space 24055.1 Definition for a metric space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

xvi CONTENTS

55.2 Definitions in other contexts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24055.3 Examples and nonexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24155.4 Relationships with compactness and completeness . . . . . . . . . . . . . . . . . . . . . . . . . . 24155.5 Use of the axiom of choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24255.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24255.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24255.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

56 Tychonoff’s theorem 24356.1 Topological definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24356.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24356.3 Proofs of Tychonoff’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24456.4 Tychonoff’s theorem and the axiom of choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24456.5 Proof of the axiom of choice from Tychonoff’s theorem . . . . . . . . . . . . . . . . . . . . . . . 24556.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24556.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

57 Union (set theory) 24757.1 Union of two sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24757.2 Algebraic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24857.3 Finite unions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24957.4 Arbitrary unions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

57.4.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24957.4.2 Union and intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

57.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25057.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25057.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

58 σ-compact space 25158.1 Properties and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25158.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25158.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25258.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25258.5 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 253

58.5.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25358.5.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26058.5.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264

Chapter 1

a-paracompact space

In mathematics, in the field of topology, a topological space is said to be a-paracompact if every open cover of thespace has a locally finite refinement. In contrast to the definition of paracompactness, the refinement is not requiredto be open.Every paracompact space is a-paracompact, and in regular spaces the two notions coincide.

1.1 References• Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.

1

Chapter 2

Binary relation

“Relation (mathematics)" redirects here. For a more general notion of relation, see finitary relation. For a morecombinatorial viewpoint, see theory of relations. For other uses, see Relation § Mathematics.

In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is asubset of the Cartesian product A2 = A × A. More generally, a binary relation between two sets A and B is a subsetof A × B. The terms correspondence, dyadic relation and 2-place relation are synonyms for binary relation.An example is the "divides" relation between the set of prime numbers P and the set of integers Z, in which everyprime p is associated with every integer z that is a multiple of p (but with no integer that is not a multiple of p). Inthis relation, for instance, the prime 2 is associated with numbers that include −4, 0, 6, 10, but not 1 or 9; and theprime 3 is associated with numbers that include 0, 6, and 9, but not 4 or 13.Binary relations are used in many branches of mathematics to model concepts like "is greater than", "is equal to", and“divides” in arithmetic, "is congruent to" in geometry, “is adjacent to” in graph theory, “is orthogonal to” in linearalgebra and many more. The concept of function is defined as a special kind of binary relation. Binary relations arealso heavily used in computer science.A binary relation is the special case n = 2 of an n-ary relation R ⊆ A1 × … × An, that is, a set of n-tuples where thejth component of each n-tuple is taken from the jth domain Aj of the relation. An example for a ternary relation onZ×Z×Z is “lies between ... and ...”, containing e.g. the triples (5,2,8), (5,8,2), and (−4,9,−7).In some systems of axiomatic set theory, relations are extended to classes, which are generalizations of sets. Thisextension is needed for, among other things, modeling the concepts of “is an element of” or “is a subset of” in settheory, without running into logical inconsistencies such as Russell’s paradox.

2.1 Formal definition

A binary relation R is usually defined as an ordered triple (X, Y, G) where X and Y are arbitrary sets (or classes), andG is a subset of the Cartesian product X × Y. The sets X and Y are called the domain (or the set of departure) andcodomain (or the set of destination), respectively, of the relation, and G is called its graph.The statement (x,y) ∈ G is read "x is R-related to y", and is denoted by xRy or R(x,y). The latter notation correspondsto viewing R as the characteristic function on X × Y for the set of pairs of G.The order of the elements in each pair ofG is important: if a ≠ b, then aRb and bRa can be true or false, independentlyof each other. Resuming the above example, the prime 3 divides the integer 9, but 9 doesn't divide 3.A relation as defined by the triple (X, Y, G) is sometimes referred to as a correspondence instead.[1] In this case therelation from X to Y is the subset G of X × Y, and “from X to Y" must always be either specified or implied by thecontext when referring to the relation. In practice correspondence and relation tend to be used interchangeably.

2

2.2. SPECIAL TYPES OF BINARY RELATIONS 3

2.1.1 Is a relation more than its graph?

According to the definition above, two relations with identical graphs but different domains or different codomainsare considered different. For example, ifG = (1, 2), (1, 3), (2, 7) , then (Z,Z, G) , (R,N, G) , and (N,R, G) arethree distinct relations, where Z is the set of integers and R is the set of real numbers.Especially in set theory, binary relations are often defined as sets of ordered pairs, identifying binary relations withtheir graphs. The domain of a binary relation R is then defined as the set of all x such that there exists at least oney such that (x, y) ∈ R , the range of R is defined as the set of all y such that there exists at least one x such that(x, y) ∈ R , and the field of R is the union of its domain and its range.[2][3][4]

A special case of this difference in points of view applies to the notion of function. Many authors insist on distin-guishing between a function’s codomain and its range. Thus, a single “rule,” like mapping every real number x tox2, can lead to distinct functions f : R → R and f : R → R+ , depending on whether the images under thatrule are understood to be reals or, more restrictively, non-negative reals. But others view functions as simply sets ofordered pairs with unique first components. This difference in perspectives does raise some nontrivial issues. As anexample, the former camp considers surjectivity—or being onto—as a property of functions, while the latter sees itas a relationship that functions may bear to sets.Either approach is adequate for most uses, provided that one attends to the necessary changes in language, notation,and the definitions of concepts like restrictions, composition, inverse relation, and so on. The choice between the twodefinitions usually matters only in very formal contexts, like category theory.

2.1.2 Example

Example: Suppose there are four objects ball, car, doll, gun and four persons John, Mary, Ian, Venus. Supposethat John owns the ball, Mary owns the doll, and Venus owns the car. Nobody owns the gun and Ian owns nothing.Then the binary relation “is owned by” is given as

R = (ball, car, doll, gun, John, Mary, Ian, Venus, (ball, John), (doll, Mary), (car, Venus)).

Thus the first element of R is the set of objects, the second is the set of persons, and the last element is a set of orderedpairs of the form (object, owner).The pair (ball, John), denoted by ₐ RJₒ means that the ball is owned by John.Two different relations could have the same graph. For example: the relation

(ball, car, doll, gun, John, Mary, Venus, (ball, John), (doll, Mary), (car, Venus))

is different from the previous one as everyone is an owner. But the graphs of the two relations are the same.Nevertheless, R is usually identified or even defined as G(R) and “an ordered pair (x, y) ∈ G(R)" is usually denoted as"(x, y) ∈ R".

2.2 Special types of binary relations

Some important types of binary relations R between two sets X and Y are listed below. To emphasize that X and Ycan be different sets, some authors call such binary relations heterogeneous.[5][6]

Uniqueness properties:

• injective (also called left-unique[7]): for all x and z in X and y in Y it holds that if xRy and zRy then x = z. Forexample, the green relation in the diagram is injective, but the red relation is not, as it relates e.g. both x = −5and z = +5 to y = 25.

• functional (also called univalent[8] or right-unique[7] or right-definite[9]): for all x in X, and y and z in Yit holds that if xRy and xRz then y = z; such a binary relation is called a partial function. Both relations inthe picture are functional. An example for a non-functional relation can be obtained by rotating the red graphclockwise by 90 degrees, i.e. by considering the relation x=y2 which relates e.g. x=25 to both y=−5 and z=+5.

4 CHAPTER 2. BINARY RELATION

Example relations between real numbers. Red: y=x2. Green: y=2x+20.

• one-to-one (also written 1-to-1): injective and functional. The green relation is one-to-one, but the red is not.

Totality properties:

• left-total:[7] for all x in X there exists a y in Y such that xRy. For example R is left-total when it is a functionor a multivalued function. Note that this property, although sometimes also referred to as total, is differentfrom the definition of total in the next section. Both relations in the picture are left-total. The relation x=y2,obtained from the above rotation, is not left-total, as it doesn't relate, e.g., x = −14 to any real number y.

• surjective (also called right-total[7] or onto): for all y in Y there exists an x in X such that xRy. The greenrelation is surjective, but the red relation is not, as it doesn't relate any real number x to e.g. y = −14.

Uniqueness and totality properties:

2.3. RELATIONS OVER A SET 5

• A function: a relation that is functional and left-total. Both the green and the red relation are functions.

• An injective function: a relation that is injective, functional, and left-total.

• A surjective function or surjection: a relation that is functional, left-total, and right-total.

• A bijection: a surjective one-to-one or surjective injective function is said to be bijective, also known asone-to-one correspondence.[10] The green relation is bijective, but the red is not.

2.2.1 Difunctional

Less commonly encountered is the notion of difunctional (or regular) relation, defined as a relation R such thatR=RR−1R.[11]

To understand this notion better, it helps to consider a relation as mapping every element x∈X to a set xR = y∈Y| xRy .[11] This set is sometimes called the successor neighborhood of x in R; one can define the predecessorneighborhood analogously.[12] Synonymous terms for these notions are afterset and respectively foreset.[5]

A difunctional relation can then be equivalently characterized as a relation R such that wherever x1R and x2R have anon-empty intersection, then these two sets coincide; formally x1R ∩ x2R ≠ ∅ implies x1R = x2R.[11]

As examples, any function or any functional (right-unique) relation is difunctional; the converse doesn't hold. If oneconsiders a relation R from set to itself (X = Y), then if R is both transitive and symmetric (i.e. a partial equivalencerelation), then it is also difunctional.[13] The converse of this latter statement also doesn't hold.A characterization of difunctional relations, which also explains their name, is to consider two functions f: A → Cand g: B→ C and then define the following set which generalizes the kernel of a single function as joint kernel: ker(f,g) = (a, b) ∈ A × B | f(a) = g(b) . Every difunctional relation R ⊆ A × B arises as the joint kernel of two functionsf: A→ C and g: B→ C for some set C.[14]

In automata theory, the term rectangular relation has also been used to denote a difunctional relation. This ter-minology is justified by the fact that when represented as a boolean matrix, the columns and rows of a difunctionalrelation can be arranged in such a way as to present rectangular blocks of true on the (asymmetric) main diagonal.[15]Other authors however use the term “rectangular” to denote any heterogeneous relation whatsoever.[6]

2.3 Relations over a set

If X = Y then we simply say that the binary relation is over X, or that it is an endorelation over X.[16] In computerscience, such a relation is also called a homogeneous (binary) relation.[16][17][6] Some types of endorelations arewidely studied in graph theory, where they are known as simple directed graphs permitting loops.The set of all binary relations Rel(X) on a set X is the set 2X × X which is a Boolean algebra augmented with theinvolution of mapping of a relation to its inverse relation. For the theoretical explanation see Relation algebra.Some important properties of a binary relation R over a set X are:

• reflexive: for all x in X it holds that xRx. For example, “greater than or equal to” (≥) is a reflexive relation but“greater than” (>) is not.

• irreflexive (or strict): for all x in X it holds that not xRx. For example, > is an irreflexive relation, but ≥ is not.

• coreflexive: for all x and y in X it holds that if xRy then x = y. An example of a coreflexive relation is therelation on integers in which each odd number is related to itself and there are no other relations. The equalityrelation is the only example of a both reflexive and coreflexive relation.

The previous 3 alternatives are far from being exhaustive; e.g. the red relation y=x2 from theabove picture is neither irreflexive, nor coreflexive, nor reflexive, since it contains the pair(0,0), and (2,4), but not (2,2), respectively.

• symmetric: for all x and y in X it holds that if xRy then yRx. “Is a blood relative of” is a symmetric relation,because x is a blood relative of y if and only if y is a blood relative of x.

6 CHAPTER 2. BINARY RELATION

• antisymmetric: for all x and y in X, if xRy and yRx then x = y. For example, ≥ is anti-symmetric (so is >, butonly because the condition in the definition is always false).[18]

• asymmetric: for all x and y in X, if xRy then not yRx. A relation is asymmetric if and only if it is bothanti-symmetric and irreflexive.[19] For example, > is asymmetric, but ≥ is not.

• transitive: for all x, y and z in X it holds that if xRy and yRz then xRz. A transitive relation is irreflexive if andonly if it is asymmetric.[20] For example, “is ancestor of” is transitive, while “is parent of” is not.

• total: for all x and y in X it holds that xRy or yRx (or both). This definition for total is different from left totalin the previous section. For example, ≥ is a total relation.

• trichotomous: for all x and y in X exactly one of xRy, yRx or x = y holds. For example, > is a trichotomousrelation, while the relation “divides” on natural numbers is not.[21]

• Euclidean: for all x, y and z in X it holds that if xRy and xRz, then yRz (and zRy). Equality is a Euclideanrelation because if x=y and x=z, then y=z.

• serial: for all x in X, there exists y in X such that xRy. "Is greater than" is a serial relation on the integers. Butit is not a serial relation on the positive integers, because there is no y in the positive integers (i.e. the naturalnumbers) such that 1>y.[22] However, "is less than" is a serial relation on the positive integers, the rationalnumbers and the real numbers. Every reflexive relation is serial: for a given x, choose y=x. A serial relation canbe equivalently characterized as every element having a non-empty successor neighborhood (see the previoussection for the definition of this notion). Similarly an inverse serial relation is a relation in which every elementhas non-empty predecessor neighborhood.[12]

• set-like (or local): for every x in X, the class of all y such that yRx is a set. (This makes sense only if relationson proper classes are allowed.) The usual ordering < on the class of ordinal numbers is set-like, while its inverse> is not.

A relation that is reflexive, symmetric, and transitive is called an equivalence relation. A relation that is symmetric,transitive, and serial is also reflexive. A relation that is only symmetric and transitive (without necessarily beingreflexive) is called a partial equivalence relation.A relation that is reflexive, antisymmetric, and transitive is called a partial order. A partial order that is total is calleda total order, simple order, linear order, or a chain.[23] A linear order where every nonempty subset has a least elementis called a well-order.

2.4 Operations on binary relations

If R, S are binary relations over X and Y, then each of the following is a binary relation over X and Y :

• Union: R ∪ S ⊆ X × Y, defined as R ∪ S = (x, y) | (x, y) ∈ R or (x, y) ∈ S . For example, ≥ is the union of >and =.

• Intersection: R ∩ S ⊆ X × Y, defined as R ∩ S = (x, y) | (x, y) ∈ R and (x, y) ∈ S .

If R is a binary relation over X and Y, and S is a binary relation over Y and Z, then the following is a binary relationover X and Z: (see main article composition of relations)

• Composition: S ∘ R, also denoted R ; S (or more ambiguously R ∘ S), defined as S ∘ R = (x, z) | there existsy ∈ Y, such that (x, y) ∈ R and (y, z) ∈ S . The order of R and S in the notation S ∘ R, used here agrees withthe standard notational order for composition of functions. For example, the composition “is mother of” ∘ “isparent of” yields “is maternal grandparent of”, while the composition “is parent of” ∘ “is mother of” yields “isgrandmother of”.

2.4. OPERATIONS ON BINARY RELATIONS 7

A relation R on sets X and Y is said to be contained in a relation S on X and Y if R is a subset of S, that is, if x R yalways implies x S y. In this case, if R and S disagree, R is also said to be smaller than S. For example, > is containedin ≥.If R is a binary relation over X and Y, then the following is a binary relation over Y and X:

• Inverse or converse: R −1, defined as R −1 = (y, x) | (x, y) ∈ R . A binary relation over a set is equal to itsinverse if and only if it is symmetric. See also duality (order theory). For example, “is less than” (<) is theinverse of “is greater than” (>).

If R is a binary relation over X, then each of the following is a binary relation over X:

• Reflexive closure: R =, defined as R = = (x, x) | x ∈ X ∪ R or the smallest reflexive relation over X containingR. This can be proven to be equal to the intersection of all reflexive relations containing R.

• Reflexive reduction: R ≠, defined as R ≠ = R \ (x, x) | x ∈ X or the largest irreflexive relation over Xcontained in R.

• Transitive closure: R +, defined as the smallest transitive relation over X containing R. This can be seen to beequal to the intersection of all transitive relations containing R.

• Transitive reduction: R −, defined as a minimal relation having the same transitive closure as R.

• Reflexive transitive closure: R *, defined as R * = (R +) =, the smallest preorder containing R.

• Reflexive transitive symmetric closure: R ≡, defined as the smallest equivalence relation over X containingR.

2.4.1 Complement

If R is a binary relation over X and Y, then the following too:

• The complement S is defined as x S y if not x R y. For example, on real numbers, ≤ is the complement of >.

The complement of the inverse is the inverse of the complement.If X = Y, the complement has the following properties:

• If a relation is symmetric, the complement is too.

• The complement of a reflexive relation is irreflexive and vice versa.

• The complement of a strict weak order is a total preorder and vice versa.

The complement of the inverse has these same properties.

2.4.2 Restriction

The restriction of a binary relation on a set X to a subset S is the set of all pairs (x, y) in the relation for which x andy are in S.If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partialorder, total order, strict weak order, total preorder (weak order), or an equivalence relation, its restrictions are too.However, the transitive closure of a restriction is a subset of the restriction of the transitive closure, i.e., in generalnot equal. For example, restricting the relation "x is parent of y" to females yields the relation "x is mother ofthe woman y"; its transitive closure doesn't relate a woman with her paternal grandmother. On the other hand, thetransitive closure of “is parent of” is “is ancestor of"; its restriction to females does relate a woman with her paternalgrandmother.

8 CHAPTER 2. BINARY RELATION

Also, the various concepts of completeness (not to be confused with being “total”) do not carry over to restrictions.For example, on the set of real numbers a property of the relation "≤" is that every non-empty subset S of R with anupper bound in R has a least upper bound (also called supremum) in R. However, for a set of rational numbers thissupremum is not necessarily rational, so the same property does not hold on the restriction of the relation "≤" to theset of rational numbers.The left-restriction (right-restriction, respectively) of a binary relation between X and Y to a subset S of its domain(codomain) is the set of all pairs (x, y) in the relation for which x (y) is an element of S.

2.4.3 Algebras, categories, and rewriting systems

Various operations on binary endorelations can be treated as giving rise to an algebraic structure, known as relationalgebra. It should not be confused with relational algebra which deals in finitary relations (and in practice also finiteand many-sorted).For heterogenous binary relations, a category of relations arises.[6]

Despite their simplicity, binary relations are at the core of an abstract computation model known as an abstractrewriting system.

2.5 Sets versus classes

Certain mathematical “relations”, such as “equal to”, “member of”, and “subset of”, cannot be understood to be binaryrelations as defined above, because their domains and codomains cannot be taken to be sets in the usual systems ofaxiomatic set theory. For example, if we try to model the general concept of “equality” as a binary relation =, wemust take the domain and codomain to be the “class of all sets”, which is not a set in the usual set theory.In most mathematical contexts, references to the relations of equality, membership and subset are harmless becausethey can be understood implicitly to be restricted to some set in the context. The usual work-around to this problemis to select a “large enough” set A, that contains all the objects of interest, and work with the restriction =A instead of=. Similarly, the “subset of” relation ⊆ needs to be restricted to have domain and codomain P(A) (the power set ofa specific set A): the resulting set relation can be denoted ⊆A. Also, the “member of” relation needs to be restrictedto have domain A and codomain P(A) to obtain a binary relation ∈A that is a set. Bertrand Russell has shown thatassuming ∈ to be defined on all sets leads to a contradiction in naive set theory.Another solution to this problem is to use a set theory with proper classes, such as NBG or Morse–Kelley set theory,and allow the domain and codomain (and so the graph) to be proper classes: in such a theory, equality, membership,and subset are binary relations without special comment. (A minor modification needs to be made to the concept ofthe ordered triple (X, Y, G), as normally a proper class cannot be a member of an ordered tuple; or of course onecan identify the function with its graph in this context.)[24] With this definition one can for instance define a functionrelation between every set and its power set.

2.6 The number of binary relations

The number of distinct binary relations on an n-element set is 2n2 (sequence A002416 in OEIS):Notes:

• The number of irreflexive relations is the same as that of reflexive relations.

• The number of strict partial orders (irreflexive transitive relations) is the same as that of partial orders.

• The number of strict weak orders is the same as that of total preorders.

• The total orders are the partial orders that are also total preorders. The number of preorders that are neithera partial order nor a total preorder is, therefore, the number of preorders, minus the number of partial orders,minus the number of total preorders, plus the number of total orders: 0, 0, 0, 3, and 85, respectively.

• the number of equivalence relations is the number of partitions, which is the Bell number.

2.7. EXAMPLES OF COMMON BINARY RELATIONS 9

The binary relations can be grouped into pairs (relation, complement), except that for n = 0 the relation is its owncomplement. The non-symmetric ones can be grouped into quadruples (relation, complement, inverse, inverse com-plement).

2.7 Examples of common binary relations

• order relations, including strict orders:

• greater than• greater than or equal to• less than• less than or equal to• divides (evenly)• is a subset of

• equivalence relations:

• equality• is parallel to (for affine spaces)• is in bijection with• isomorphy

• dependency relation, a finite, symmetric, reflexive relation.

• independency relation, a symmetric, irreflexive relation which is the complement of some dependency relation.

2.8 See also

• Confluence (term rewriting)

• Hasse diagram

• Incidence structure

• Logic of relatives

• Order theory

• Triadic relation

2.9 Notes[1] Encyclopedic dictionary of Mathematics. MIT. 2000. pp. 1330–1331. ISBN 0-262-59020-4.

[2] Suppes, Patrick (1972) [originally published by D. van Nostrand Company in 1960]. Axiomatic Set Theory. Dover. ISBN0-486-61630-4.

[3] Smullyan, Raymond M.; Fitting, Melvin (2010) [revised and corrected republication of the work originally published in1996 by Oxford University Press, New York]. Set Theory and the Continuum Problem. Dover. ISBN 978-0-486-47484-7.

[4] Levy, Azriel (2002) [republication of the work published by Springer-Verlag, Berlin, Heidelberg and New York in 1979].Basic Set Theory. Dover. ISBN 0-486-42079-5.

[5] Christodoulos A. Floudas; PanosM. Pardalos (2008). Encyclopedia of Optimization (2nd ed.). Springer Science&BusinessMedia. pp. 299–300. ISBN 978-0-387-74758-3.

10 CHAPTER 2. BINARY RELATION

[6] Michael Winter (2007). Goguen Categories: A Categorical Approach to L-fuzzy Relations. Springer. pp. x–xi. ISBN978-1-4020-6164-6.

[7] Kilp, Knauer and Mikhalev: p. 3. The same four definitions appear in the following:

• Peter J. Pahl; Rudolf Damrath (2001). Mathematical Foundations of Computational Engineering: A Handbook.Springer Science & Business Media. p. 506. ISBN 978-3-540-67995-0.

• Eike Best (1996). Semantics of Sequential and Parallel Programs. Prentice Hall. pp. 19–21. ISBN 978-0-13-460643-9.

• Robert-Christoph Riemann (1999). Modelling of Concurrent Systems: Structural and Semantical Methods in the HighLevel Petri Net Calculus. Herbert Utz Verlag. pp. 21–22. ISBN 978-3-89675-629-9.

[8] Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, ISBN 978-0-521-76268-7, Chapt. 5

[9] Mäs, Stephan (2007), “Reasoning on Spatial Semantic Integrity Constraints”, Spatial Information Theory: 8th InternationalConference, COSIT 2007, Melbourne, Australiia, September 19–23, 2007, Proceedings, Lecture Notes in Computer Science4736, Springer, pp. 285–302, doi:10.1007/978-3-540-74788-8_18

[10] Note that the use of “correspondence” here is narrower than as general synonym for binary relation.

[11] Chris Brink; Wolfram Kahl; Gunther Schmidt (1997). Relational Methods in Computer Science. Springer Science &Business Media. p. 200. ISBN 978-3-211-82971-4.

[12] Yao, Y. (2004). “Semantics of Fuzzy Sets in Rough Set Theory”. Transactions on Rough Sets II. Lecture Notes in ComputerScience 3135. p. 297. doi:10.1007/978-3-540-27778-1_15. ISBN 978-3-540-23990-1.

[13] William Craig (2006). Semigroups Underlying First-order Logic. American Mathematical Soc. p. 72. ISBN 978-0-8218-6588-0.

[14] Gumm, H. P.; Zarrad, M. (2014). “Coalgebraic Simulations and Congruences”. Coalgebraic Methods in Computer Science.Lecture Notes in Computer Science 8446. p. 118. doi:10.1007/978-3-662-44124-4_7. ISBN 978-3-662-44123-7.

[15] Julius Richard Büchi (1989). Finite Automata, Their Algebras and Grammars: Towards a Theory of Formal Expressions.Springer Science & Business Media. pp. 35–37. ISBN 978-1-4613-8853-1.

[16] M. E. Müller (2012). Relational Knowledge Discovery. Cambridge University Press. p. 22. ISBN 978-0-521-19021-3.

[17] Peter J. Pahl; Rudolf Damrath (2001). Mathematical Foundations of Computational Engineering: A Handbook. SpringerScience & Business Media. p. 496. ISBN 978-3-540-67995-0.

[18] Smith, Douglas; Eggen, Maurice; St. Andre, Richard (2006),ATransition to AdvancedMathematics (6th ed.), Brooks/Cole,p. 160, ISBN 0-534-39900-2

[19] Nievergelt, Yves (2002), Foundations of Logic and Mathematics: Applications to Computer Science and Cryptography,Springer-Verlag, p. 158.

[20] Flaška, V.; Ježek, J.; Kepka, T.; Kortelainen, J. (2007). Transitive Closures of Binary Relations I (PDF). Prague: Schoolof Mathematics – Physics Charles University. p. 1. Lemma 1.1 (iv). This source refers to asymmetric relations as “strictlyantisymmetric”.

[21] Since neither 5 divides 3, nor 3 divides 5, nor 3=5.

[22] Yao, Y.Y.; Wong, S.K.M. (1995). “Generalization of rough sets using relationships between attribute values” (PDF).Proceedings of the 2nd Annual Joint Conference on Information Sciences: 30–33..

[23] Joseph G. Rosenstein, Linear orderings, Academic Press, 1982, ISBN 0-12-597680-1, p. 4

[24] Tarski, Alfred; Givant, Steven (1987). A formalization of set theory without variables. American Mathematical Society. p.3. ISBN 0-8218-1041-3.

2.10 References• M. Kilp, U. Knauer, A.V. Mikhalev, Monoids, Acts and Categories: with Applications to Wreath Products andGraphs, De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, ISBN 3-11-015248-7.

• Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, ISBN 978-0-521-76268-7.

2.11. EXTERNAL LINKS 11

2.11 External links• Hazewinkel, Michiel, ed. (2001), “Binary relation”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Chapter 3

Closed set

This article is about the complement of an open set. For a set closed under an operation, see closure (mathematics).For other uses, see Closed (disambiguation).

In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an openset.[1][2] In a topological space, a closed set can be defined as a set which contains all its limit points. In a completemetric space, a closed set is a set which is closed under the limit operation.

3.1 Equivalent definitions of a closed set

In a topological space, a set is closed if and only if it coincides with its closure. Equivalently, a set is closed if andonly if it contains all of its limit points.This is not to be confused with a closed manifold.

3.2 Properties of closed sets

A closed set contains its own boundary. In other words, if you are “outside” a closed set, you may move a smallamount in any direction and still stay outside the set. Note that this is also true if the boundary is the empty set, e.g.in the metric space of rational numbers, for the set of numbers of which the square is less than 2.

• Any intersection of closed sets is closed (including intersections of infinitely many closed sets)

• The union of finitely many closed sets is closed.

• The empty set is closed.

• The whole set is closed.

In fact, given a set X and a collection F of subsets of X that has these properties, then F will be the collection ofclosed sets for a unique topology on X. The intersection property also allows one to define the closure of a set A in aspace X, which is defined as the smallest closed subset of X that is a superset of A. Specifically, the closure of A canbe constructed as the intersection of all of these closed supersets.Sets that can be constructed as the union of countably many closed sets are denoted Fσ sets. These sets need not beclosed.

3.3 Examples of closed sets• The closed interval [a,b] of real numbers is closed. (See Interval (mathematics) for an explanation of thebracket and parenthesis set notation.)

12

3.4. MORE ABOUT CLOSED SETS 13

• The unit interval [0,1] is closed in the metric space of real numbers, and the set [0,1] ∩ Q of rational numbersbetween 0 and 1 (inclusive) is closed in the space of rational numbers, but [0,1] ∩ Q is not closed in the realnumbers.

• Some sets are neither open nor closed, for instance the half-open interval [0,1) in the real numbers.• Some sets are both open and closed and are called clopen sets.• Half-interval [1, +∞) is closed.• The Cantor set is an unusual closed set in the sense that it consists entirely of boundary points and is nowheredense.

• Singleton points (and thus finite sets) are closed in Hausdorff spaces.• If X and Y are topological spaces, a function f from X into Y is continuous if and only if preimages of closedsets in Y are closed in X.

3.4 More about closed sets

In point set topology, a set A is closed if it contains all its boundary points.The notion of closed set is defined above in terms of open sets, a concept that makes sense for topological spaces,as well as for other spaces that carry topological structures, such as metric spaces, differentiable manifolds, uniformspaces, and gauge spaces.An alternative characterization of closed sets is available via sequences and nets. A subset A of a topological spaceX is closed in X if and only if every limit of every net of elements of A also belongs to A. In a first-countable space(such as a metric space), it is enough to consider only convergent sequences, instead of all nets. One value of thischaracterisation is that it may be used as a definition in the context of convergence spaces, which are more generalthan topological spaces. Notice that this characterisation also depends on the surrounding space X, because whetheror not a sequence or net converges in X depends on what points are present in X.Whether a set is closed depends on the space in which it is embedded. However, the compact Hausdorff spacesare "absolutely closed", in the sense that, if you embed a compact Hausdorff space K in an arbitrary Hausdorffspace X, then K will always be a closed subset of X; the “surrounding space” does not matter here. Stone-Čechcompactification, a process that turns a completely regular Hausdorff space into a compact Hausdorff space, may bedescribed as adjoining limits of certain nonconvergent nets to the space.Furthermore, every closed subset of a compact space is compact, and every compact subspace of a Hausdorff spaceis closed.Closed sets also give a useful characterization of compactness: a topological space X is compact if and only ifevery collection of nonempty closed subsets of X with empty intersection admits a finite subcollection with emptyintersection.A topological space X is disconnected if there exist disjoint, nonempty, closed subsets A and B of X whose union isX. Furthermore, X is totally disconnected if it has an open basis consisting of closed sets.

3.5 See also• Open set• Clopen set• Neighbourhood

3.6 References[1] Rudin, Walter (1976). Principles of Mathematical Analysis. McGraw-Hill. ISBN 0-07-054235-X.

[2] Munkres, James R. (2000). Topology (2nd ed.). Prentice Hall. ISBN 0-13-181629-2.

Chapter 4

Closure (topology)

For other uses, see Closure (disambiguation).

In mathematics, the closure of a subset S in a topological space consists of all points in S plus the limit points of S.The closure of S is also defined as the union of S and its boundary. Intuitively, these are all the points in S and “near”S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to thenotion of interior.

4.1 Definitions

4.1.1 Point of closure

For S a subset of a Euclidean space, x is a point of closure of S if every open ball centered at x contains a point of S(this point may be x itself).This definition generalises to any subset S of a metric space X. Fully expressed, for X a metric space with metric d, xis a point of closure of S if for every r > 0, there is a y in S such that the distance d(x, y) < r. (Again, we may have x= y.) Another way to express this is to say that x is a point of closure of S if the distance d(x, S) := infd(x, s) : s inS = 0.This definition generalises to topological spaces by replacing “open ball” or “ball” with "neighbourhood". Let S bea subset of a topological space X. Then x is a point of closure (or adherent point) of S if every neighbourhood of xcontains a point of S.[1] Note that this definition does not depend upon whether neighbourhoods are required to beopen.

4.1.2 Limit point

The definition of a point of closure is closely related to the definition of a limit point. The difference between thetwo definitions is subtle but important — namely, in the definition of limit point, every neighborhood of the point xin question must contain a point of the set other than x itself.Thus, every limit point is a point of closure, but not every point of closure is a limit point. A point of closure whichis not a limit point is an isolated point. In other words, a point x is an isolated point of S if it is an element of S andif there is a neighbourhood of x which contains no other points of S other than x itself.[2]

For a given set S and point x, x is a point of closure of S if and only if x is an element of S or x is a limit point of S(or both).

4.1.3 Closure of a set

See also: Closure (mathematics)

14

4.2. EXAMPLES 15

The closure of a set S is the set of all points of closure of S, that is, the set S together with all of its limit points.[3]The closure of S is denoted cl(S), Cl(S), S or S− . The closure of a set has the following properties.[4]

• cl(S) is a closed superset of S.

• cl(S) is the intersection of all closed sets containing S.

• cl(S) is the smallest closed set containing S.

• cl(S) is the union of S and its boundary ∂(S).

• A set S is closed if and only if S = cl(S).

• If S is a subset of T, then cl(S) is a subset of cl(T).

• If A is a closed set, then A contains S if and only if A contains cl(S).

Sometimes the second or third property above is taken as the definition of the topological closure, which still makesense when applied to other types of closures (see below).[5]

In a first-countable space (such as a metric space), cl(S) is the set of all limits of all convergent sequences of pointsin S. For a general topological space, this statement remains true if one replaces “sequence” by "net" or "filter".Note that these properties are also satisfied if “closure”, “superset”, “intersection”, “contains/containing”, “smallest”and “closed” are replaced by “interior”, “subset”, “union”, “contained in”, “largest”, and “open”. For more on thismatter, see closure operator below.

4.2 Examples

Consider a sphere in 3 dimensions. Implicitly there are two regions of interest created by this sphere; the sphere itselfand its interior (which is called an open 3-ball). It is useful to be able to distinguish between the interior of 3-ball andthe surface, so we distinguish between the open 3-ball, and the closed 3-ball - the closure of the 3-ball. The closureof the open 3-ball is the open 3-ball plus the surface.In topological space:

• In any space, ∅ = cl(∅) .

• In any space X, X = cl(X).

Giving R and C the standard (metric) topology:

• If X is the Euclidean space R of real numbers, then cl((0, 1)) = [0, 1].

• If X is the Euclidean space R, then the closure of the set Q of rational numbers is the whole space R. We saythat Q is dense in R.

• If X is the complex plane C = R2, then cl(z in C : |z| > 1) = z in C : |z| ≥ 1.

• If S is a finite subset of a Euclidean space, then cl(S) = S. (For a general topological space, this property isequivalent to the T1 axiom.)

On the set of real numbers one can put other topologies rather than the standard one.

• If X = R, where R has the lower limit topology, then cl((0, 1)) = [0, 1).

• If one considers on R the discrete topology in which every set is closed (open), then cl((0, 1)) = (0, 1).

• If one considers on R the trivial topology in which the only closed (open) sets are the empty set and R itself,then cl((0, 1)) = R.

16 CHAPTER 4. CLOSURE (TOPOLOGY)

These examples show that the closure of a set depends upon the topology of the underlying space. The last twoexamples are special cases of the following.

• In any discrete space, since every set is closed (and also open), every set is equal to its closure.• In any indiscrete space X, since the only closed sets are the empty set and X itself, we have that the closureof the empty set is the empty set, and for every non-empty subset A of X, cl(A) = X. In other words, everynon-empty subset of an indiscrete space is dense.

The closure of a set also depends upon in which space we are taking the closure. For example, if X is the set ofrational numbers, with the usual relative topology induced by the Euclidean space R, and if S = q in Q : q2 > 2, q >0, then S is closed in Q, and the closure of S in Q is S; however, the closure of S in the Euclidean space R is the setof all real numbers greater than or equal to

√2.

4.3 Closure operator

See also: Closure operator

A closure operator on a set X is a mapping of the power set of X, P(X) , into itself which satisfies the Kuratowskiclosure axioms.Given a topological space (X, T ) , the mapping − : S → S− for all S ⊆ X is a closure operator on X. Conversely, if cis a closure operator on a set X, a topological space is obtained by defining the sets S with c(S) = S as closed sets (sotheir complements are the open sets of the topology).[6]

The closure operator − is dual to the interior operator o, in the sense that

S− = X \ (X \ S)o

and also

So = X \ (X \ S)−

where X denotes the underlying set of the topological space containing S, and the backslash refers to the set-theoreticdifference.Therefore, the abstract theory of closure operators and the Kuratowski closure axioms can be easily translated intothe language of interior operators, by replacing sets with their complements.

4.4 Facts about closures

The set S is closed if and only if Cl(S) = S . In particular:

• The closure of the empty set is the empty set;• The closure of X itself is X .• The closure of an intersection of sets is always a subset of (but need not be equal to) the intersection of theclosures of the sets.

• In a union of finitely many sets, the closure of the union and the union of the closures are equal; the union ofzero sets is the empty set, and so this statement contains the earlier statement about the closure of the emptyset as a special case.

• The closure of the union of infinitely many sets need not equal the union of the closures, but it is always asuperset of the union of the closures.

If A is a subspace ofX containing S , then the closure of S computed in A is equal to the intersection of A and theclosure of S computed inX : ClA(S) = A ∩ClX(S) . In particular, S is dense in A if and only if A is a subset ofClX(S) .

4.5. CATEGORICAL INTERPRETATION 17

4.5 Categorical interpretation

One may elegantly define the closure operator in terms of universal arrows, as follows.The powerset of a set X may be realized as a partial order category P in which the objects are subsets and themorphisms are inclusions A → B whenever A is a subset of B. Furthermore, a topology T on X is a subcategory ofP with inclusion functor I : T → P . The set of closed subsets containing a fixed subset A ⊆ X can be identifiedwith the comma category (A ↓ I) . This category — also a partial order — then has initial object Cl(A). Thus thereis a universal arrow from A to I, given by the inclusion A→ Cl(A) .Similarly, since every closed set containing X \ A corresponds with an open set contained in A we can interpret thecategory (I ↓ X \A) as the set of open subsets contained in A, with terminal object int(A) , the interior of A.All properties of the closure can be derived from this definition and a few properties of the above categories. More-over, this definition makes precise the analogy between the topological closure and other types of closures (forexample algebraic), since all are examples of universal arrows.

4.6 See also• Closure algebra

4.7 Notes[1] Schubert, p. 20

[2] Kuratowski, p. 75

[3] Hocking Young, p. 4

[4] Croom, p. 104

[5] Gemignani, p. 55, Pervin, p. 40 and Baker, p. 38 use the second property as the definition.

[6] Pervin, p. 41

4.8 References• Baker, Crump W. (1991), Introduction to Topology, Wm. C. Brown Publisher, ISBN 0-697-05972-3

• Croom, Fred H. (1989), Principles of Topology, Saunders College Publishing, ISBN 0-03-012813-7

• Gemignani, Michael C. (1990) [1967], Elementary Topology (2nd ed.), Dover, ISBN 0-486-66522-4

• Hocking, John G.; Young, Gail S. (1988) [1961], Topology, Dover, ISBN 0-486-65676-4

• Kuratowski, K. (1966), Topology I, Academic Press

• Pervin, William J. (1965), Foundations of General Topology, Academic Press

• Schubert, Horst (1968), Topology, Allyn and Bacon

4.9 External links• Hazewinkel, Michiel, ed. (2001), “Closure of a set”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Chapter 5

Compact operator

In functional analysis, a branch of mathematics, a compact operator is a linear operator L from a Banach space Xto another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset ofY. Such an operator is necessarily a bounded operator, and so continuous.Any bounded operator L that has finite rank is a compact operator; indeed, the class of compact operators is a naturalgeneralisation of the class of finite-rank operators in an infinite-dimensional setting. When Y is a Hilbert space, itis true that any compact operator is a limit of finite-rank operators, so that the class of compact operators can bedefined alternatively as the closure in the operator norm of the finite-rank operators. Whether this was true in generalfor Banach spaces (the approximation property) was an unsolved question for many years; in the end Per Enflo gavea counter-example.The origin of the theory of compact operators is in the theory of integral equations, where integral operators supplyconcrete examples of such operators. A typical Fredholm integral equation gives rise to a compact operator K onfunction spaces; the compactness property is shown by equicontinuity. The method of approximation by finite-rankoperators is basic in the numerical solution of such equations. The abstract idea of Fredholm operator is derived fromthis connection.

5.1 Equivalent formulations

A bounded operator T : X → Y is compact if and only if any of the following is true

• Image of the unit ball in X under T is relatively compact in Y.

• Image of any bounded set under T is relatively compact in Y.

• Image of any bounded set under T is totally bounded in Y.

• there exists a neighbourhood of 0, U ⊂ X , and compact set V ⊂ Y such that T (U) ⊂ V .

• For any sequence (xn)n∈N from the unit ball in X, the sequence (Txn)n∈N contains a Cauchy subsequence.

Note that if a linear operator is compact, then it is easy to see that it is bounded, and hence continuous.

5.2 Important properties

In the following, X, Y, Z, W are Banach spaces, B(X, Y) is the space of bounded operators from X to Y with theoperator norm, K(X, Y) is the space of compact operators from X to Y, B(X) = B(X, X), K(X) = K(X, X), idX is theidentity operator on X.

• K(X, Y) is a closed subspace of B(X, Y): Let Tn, n ∈ N, be a sequence of compact operators from one Banachspace to the other, and suppose that Tn converges to T with respect to the operator norm. Then T is alsocompact.

18

5.3. ORIGINS IN INTEGRAL EQUATION THEORY 19

• Conversely, if X, Y are Hilbert spaces, then every compact operator from X to Y is the limit of finite rankoperators. Notably, this is false for general Banach spaces X and Y.

• B(Y,Z) K(X,Y ) B(W,X) ⊆ K(W,Z). In particular, K(X) forms a two-sided operator ideal in B(X).

• idX is compact if and only if X has finite dimension.

• For any T ∈ K(X), idX − T is a Fredholm operator of index 0. In particular, im (idX − T ) is closed. Thisis essential in developing the spectral properties of compact operators. One can notice the similarity betweenthis property and the fact that, if M and N are subspaces of a Banach space where M is closed and N isfinite-dimensional, then M + N is also closed.

• Any compact operator is strictly singular, but not vice versa.[1]

• An operator is compact if and only if its adjoint is compact (Schauder’s theorem).

5.3 Origins in integral equation theory

A crucial property of compact operators is the Fredholm alternative, which asserts that the existence of solution oflinear equations of the form(λK + I)u = f

(where K is a compact operator, f is a given function, and u is the unknown function to be solved for) behaves muchlike as in finite dimensions. The spectral theory of compact operators then follows, and it is due to Frigyes Riesz(1918). It shows that a compact operator K on an infinite-dimensional Banach space has spectrum that is either afinite subset of C which includes 0, or the spectrum is a countably infinite subset of C which has 0 as its only limitpoint. Moreover, in either case the non-zero elements of the spectrum are eigenvalues of K with finite multiplicities(so that K − λI has a finite-dimensional kernel for all complex λ ≠ 0).An important example of a compact operator is compact embedding of Sobolev spaces, which, along with the Gårdinginequality and the Lax–Milgram theorem, can be used to convert an elliptic boundary value problem into a Fredholmintegral equation.[2] Existence of the solution and spectral properties then follow from the theory of compact opera-tors; in particular, an elliptic boundary value problem on a bounded domain has infinitely many isolated eigenvalues.One consequence is that a solid body can vibrate only at isolated frequencies, given by the eigenvalues, and arbitrarilyhigh vibration frequencies always exist.The compact operators from a Banach space to itself form a two-sided ideal in the algebra of all bounded operatorson the space. Indeed, the compact operators on an infinite-dimensional Hilbert space form a maximal ideal, so thequotient algebra, known as the Calkin algebra, is simple.

5.4 Compact operator on Hilbert spaces

Main article: Compact operator on Hilbert space

An equivalent definition of compact operators on a Hilbert space may be given as follows.An operator T on an infinite-dimensional Hilbert spaceH

T : H → H

is said to be compact if it can be written in the form

T =∞∑

n=1

λn⟨fn, ·⟩gn ,

20 CHAPTER 5. COMPACT OPERATOR

where f1, f2, . . . and g1, g2, . . . are (not necessarily complete) orthonormal sets, and λ1, λ2, . . . is a sequence ofpositive numbers with limit zero, called the singular values of the operator. The singular values can accumulate onlyat zero. If the sequence becomes stationary at zero, that is λN+k = 0 for some N ∈ N, and every k = 1, 2, . . . ,then the operator has finite rank, i.e, a finite-dimensional range and can be written as

T =N∑

n=1

λn⟨fn, ·⟩gn .

The bracket ⟨·, ·⟩ is the scalar product on the Hilbert space; the sum on the right hand side converges in the operatornorm.An important subclass of compact operators is the trace-class or nuclear operators.

5.5 Completely continuous operators

Let X and Y be Banach spaces. A bounded linear operator T : X→ Y is called completely continuous if, for everyweakly convergent sequence (xn) from X, the sequence (Txn) is norm-convergent in Y (Conway 1985, §VI.3).Compact operators on a Banach space are always completely continuous. If X is a reflexive Banach space, then everycompletely continuous operator T : X→ Y is compact.

5.6 Examples• Every finite rank operator is compact.

• For ℓp and a sequence (tn) converging to zero, the multiplication operator (Tx)n = tn xn is compact.

• For some fixed g ∈ C([0, 1]; R), define the linear operator T from C([0, 1]; R) to C([0, 1]; R) by

(Tf)(x) =

∫ x

0

f(t)g(t) dt.

That the operator T is indeed compact follows from the Ascoli theorem.

• More generally, if Ω is any domain in Rn and the integral kernel k : Ω × Ω → R is a Hilbert—Schmidt kernel,then the operator T on L2(Ω; R) defined by

(Tf)(x) =

∫Ω

k(x, y)f(y) dy

is a compact operator.

• By Riesz’s lemma, the identity operator is a compact operator if and only if the space is finite-dimensional.

5.7 See also• Spectral theory of compact operators• Fredholm operator• Fredholm integral equations• Fredholm alternative• Compact embedding• Strictly singular operator

5.8. NOTES 21

5.8 Notes[1] N.L. Carothers, A Short Course on Banach Space Theory, (2005) London Mathematical Society Student Texts 64, Cam-

bridge University Press.

[2] William McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000

5.9 References• Conway, John B. (1985). A course in functional analysis. Springer-Verlag. ISBN 3-540-96042-2.

• Renardy, Michael and Rogers, Robert C. (2004). An introduction to partial differential equations. Texts inApplied Mathematics 13 (Second ed.). New York: Springer-Verlag. p. 356. ISBN 0-387-00444-0. (Section7.5)

• Kutateladze, S.S. (1996). Fundamentals of Functional Analysis. Texts in Mathematical Sciences 12 (Seconded.). New York: Springer-Verlag. p. 292. ISBN 978-0-7923-3898-7.

Chapter 6

Compact space

“Compactness” redirects here. For the concept in first-order logic, see Compactness theorem.In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of

The interval A = (-∞, −2] is not compact because it is not bounded. The interval C = (2, 4) is not compact because it is not closed.The interval B = [0, 1] is compact because it is both closed and bounded.

a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all itspoints lie within some fixed distance of each other). Examples include a closed interval, a rectangle, or a finite set ofpoints. This notion is defined for more general topological spaces than Euclidean space in various ways.One such generalization is that a space is sequentially compact if any infinite sequence of points sampled from thespace must frequently (infinitely often) get arbitrarily close to some point of the space. An equivalent definition isthat every sequence of points must have an infinite subsequence that converges to some point of the space. TheHeine-Borel theorem states that a subset of Euclidean space is compact in this sequential sense if and only if it isclosed and bounded. Thus, if one chooses an infinite number of points in the closed unit interval [0, 1] some of thosepoints must get arbitrarily close to some real number in that space. For instance, some of the numbers 1/2, 4/5, 1/3,5/6, 1/4, 6/7, … accumulate to 0 (others accumulate to 1). The same set of points would not accumulate to any pointof the open unit interval (0, 1); so the open unit interval is not compact. Euclidean space itself is not compact sinceit is not bounded. In particular, the sequence of points 0, 1, 2, 3, … has no subsequence that converges to any givenreal number.Apart from closed and bounded subsets of Euclidean space, typical examples of compact spaces include spacesconsisting not of geometrical points but of functions. The term compact was introduced into mathematics by MauriceFréchet in 1904 as a distillation of this concept. Compactness in this more general situation plays an extremelyimportant role in mathematical analysis, because many classical and important theorems of 19th century analysis,such as the extreme value theorem, are easily generalized to this situation. A typical application is furnished by the

22

6.1. HISTORICAL DEVELOPMENT 23

Arzelà–Ascoli theorem or the Peano existence theorem, in which one is able to conclude the existence of a functionwith some required properties as a limiting case of some more elementary construction.Various equivalent notions of compactness, including sequential compactness and limit point compactness, can bedeveloped in general metric spaces. In general topological spaces, however, different notions of compactness are notnecessarily equivalent. The most useful notion, which is the standard definition of the unqualified term compactness,is phrased in terms of the existence of finite families of open sets that "cover" the space in the sense that each pointof the space must lie in some set contained in the family. This more subtle notion, introduced by Pavel Alexandrovand Pavel Urysohn in 1929, exhibits compact spaces as generalizations of finite sets. In spaces that are compact inthis sense, it is often possible to patch together information that holds locally—that is, in a neighborhood of eachpoint—into corresponding statements that hold throughout the space, and many theorems are of this character.The term compact set is sometimes a synonym for compact space, but usually refers to a compact subspace of atopological space.

6.1 Historical development

In the 19th century, several disparate mathematical properties were understood that would later be seen as conse-quences of compactness. On the one hand, Bernard Bolzano (1817) had been aware that any bounded sequence ofpoints (in the line or plane, for instance) has a subsequence that must eventually get arbitrarily close to some otherpoint, called a limit point. Bolzano’s proof relied on the method of bisection: the sequence was placed into an intervalthat was then divided into two equal parts, and a part containing infinitely many terms of the sequence was selected.The process could then be repeated by dividing the resulting smaller interval into smaller and smaller parts until itcloses down on the desired limit point. The full significance of Bolzano’s theorem, and its method of proof, wouldnot emerge until almost 50 years later when it was rediscovered by Karl Weierstrass.[1]

In the 1880s, it became clear that results similar to the Bolzano–Weierstrass theorem could be formulated for spacesof functions rather than just numbers or geometrical points. The idea of regarding functions as themselves pointsof a generalized space dates back to the investigations of Giulio Ascoli and Cesare Arzelà.[2] The culmination oftheir investigations, the Arzelà–Ascoli theorem, was a generalization of the Bolzano–Weierstrass theorem to familiesof continuous functions, the precise conclusion of which was that it was possible to extract a uniformly convergentsequence of functions from a suitable family of functions. The uniform limit of this sequence then played preciselythe same role as Bolzano’s “limit point”. Towards the beginning of the twentieth century, results similar to that ofArzelà and Ascoli began to accumulate in the area of integral equations, as investigated by David Hilbert and ErhardSchmidt. For a certain class of Green functions coming from solutions of integral equations, Schmidt had shown thata property analogous to the Arzelà–Ascoli theorem held in the sense of mean convergence—or convergence in whatwould later be dubbed a Hilbert space. This ultimately led to the notion of a compact operator as an offshoot of thegeneral notion of a compact space. It was Maurice Fréchet who, in 1906, had distilled the essence of the Bolzano–Weierstrass property and coined the term compactness to refer to this general phenomenon (he used the term alreadyin his 1904 paper[3] which led to the famous 1906 thesis) .However, a different notion of compactness altogether had also slowly emerged at the end of the 19th century fromthe study of the continuum, which was seen as fundamental for the rigorous formulation of analysis. In 1870, EduardHeine showed that a continuous function defined on a closed and bounded interval was in fact uniformly continuous.In the course of the proof, he made use of a lemma that from any countable cover of the interval by smaller openintervals, it was possible to select a finite number of these that also covered it. The significance of this lemma wasrecognized by Émile Borel (1895), and it was generalized to arbitrary collections of intervals by Pierre Cousin (1895)and Henri Lebesgue (1904). The Heine–Borel theorem, as the result is now known, is another special propertypossessed by closed and bounded sets of real numbers.This property was significant because it allowed for the passage from local information about a set (such as thecontinuity of a function) to global information about the set (such as the uniform continuity of a function). Thissentiment was expressed by Lebesgue (1904), who also exploited it in the development of the integral now bearinghis name. Ultimately the Russian school of point-set topology, under the direction of Pavel Alexandrov and PavelUrysohn, formulated Heine–Borel compactness in a way that could be applied to the modern notion of a topologicalspace. Alexandrov & Urysohn (1929) showed that the earlier version of compactness due to Fréchet, now called(relative) sequential compactness, under appropriate conditions followed from the version of compactness that wasformulated in terms of the existence of finite subcovers. It was this notion of compactness that became the dominantone, because it was not only a stronger property, but it could be formulated in a more general setting with a minimumof additional technical machinery, as it relied only on the structure of the open sets in a space.

24 CHAPTER 6. COMPACT SPACE

6.2 Basic examples

An example of a compact space is the (closed) unit interval [0,1] of real numbers. If one chooses an infinite numberof distinct points in the unit interval, then there must be some accumulation point in that interval. For instance,the odd-numbered terms of the sequence 1, 1/2, 1/3, 3/4, 1/5, 5/6, 1/7, 7/8, … get arbitrarily close to 0, while theeven-numbered ones get arbitrarily close to 1. The given example sequence shows the importance of including theboundary points of the interval, since the limit points must be in the space itself — an open (or half-open) interval ofthe real numbers is not compact. It is also crucial that the interval be bounded, since in the interval [0,∞) one couldchoose the sequence of points 0, 1, 2, 3, …, of which no sub-sequence ultimately gets arbitrarily close to any givenreal number.In two dimensions, closed disks are compact since for any infinite number of points sampled from a disk, some subsetof those points must get arbitrarily close either to a point within the disc, or to a point on the boundary. However, anopen disk is not compact, because a sequence of points can tend to the boundary without getting arbitrarily close toany point in the interior. Likewise, spheres are compact, but a sphere missing a point is not since a sequence of pointscan tend to the missing point, thereby not getting arbitrarily close to any point within the space. Lines and planes arenot compact, since one can take a set of equally-spaced points in any given direction without approaching any point.

6.3 Definitions

Various definitions of compactness may apply, depending on the level of generality. A subset of Euclidean space inparticular is called compact if it is closed and bounded. This implies, by the Bolzano–Weierstrass theorem, that anyinfinite sequence from the set has a subsequence that converges to a point in the set. Various equivalent notions ofcompactness, such as sequential compactness and limit point compactness, can be developed in general metric spaces.In general topological spaces, however, the different notions of compactness are not equivalent, and the most usefulnotion of compactness—originally called bicompactness—is defined using covers consisting of open sets (see Opencover definition below). That this form of compactness holds for closed and bounded subsets of Euclidean space isknown as the Heine–Borel theorem. Compactness, when defined in this manner, often allows one to take informationthat is known locally—in a neighbourhood of each point of the space—and to extend it to information that holdsglobally throughout the space. An example of this phenomenon is Dirichlet’s theorem, to which it was originallyapplied by Heine, that a continuous function on a compact interval is uniformly continuous; here, continuity is a localproperty of the function, and uniform continuity the corresponding global property.

6.3.1 Open cover definition

Formally, a topological space X is called compact if each of its open covers has a finite subcover. Otherwise, it iscalled non-compact. Explicitly, this means that for every arbitrary collection

Uαα∈A

of open subsets of X such that

X =∪α∈A

Uα,

there is a finite subset J of A such that

X =∪i∈J

Ui.

Some branches of mathematics such as algebraic geometry, typically influenced by the French school of Bourbaki,use the term quasi-compact for the general notion, and reserve the term compact for topological spaces that are bothHausdorff and quasi-compact. A compact set is sometimes referred to as a compactum, plural compacta.

6.3. DEFINITIONS 25

6.3.2 Equivalent definitions

Assuming the axiom of choice, the following are equivalent:

1. A topological space X is compact.

2. Every open cover of X has a finite subcover.

3. X has a sub-base such that every cover of the space bymembers of the sub-base has a finite subcover (Alexander’ssub-base theorem)

4. Any collection of closed subsets of X with the finite intersection property has nonempty intersection.

5. Every net on X has a convergent subnet (see the article on nets for a proof).

6. Every filter on X has a convergent refinement.

7. Every ultrafilter on X converges to at least one point.

8. Every infinite subset of X has a complete accumulation point.[4]

Euclidean space

For any subset A of Euclidean space Rn, A is compact if and only if it is closed and bounded; this is the Heine–Boreltheorem.As a Euclidean space is a metric space, the conditions in the next subsection also apply to all of its subsets. Of allof the equivalent conditions, it is in practice easiest to verify that a subset is closed and bounded, for example, for aclosed interval or closed n-ball.

Metric spaces

For any metric space (X,d), the following are equivalent:

1. (X,d) is compact.

2. (X,d) is complete and totally bounded (this is also equivalent to compactness for uniform spaces).[5]

3. (X,d) is sequentially compact; that is, every sequence in X has a convergent subsequence whose limit is in X(this is also equivalent to compactness for first-countable uniform spaces).

4. (X,d) is limit point compact; that is, every infinite subset of X has at least one limit point in X.

5. (X,d) is an image of a continuous function from the Cantor set.[6]

A compact metric space (X,d) also satisfies the following properties:

1. Lebesgue’s number lemma: For every open cover of X, there exists a number δ > 0 such that every subset ofX of diameter < δ is contained in some member of the cover.

2. (X,d) is second-countable, separable and Lindelöf – these three conditions are equivalent for metric spaces.The converse is not true; e.g., a countable discrete space satisfies these three conditions, but is not compact.

3. X is closed and bounded (as a subset of any metric space whose restricted metric is d). The converse may failfor a non-Euclidean space; e.g. the real line equipped with the discrete topology is closed and bounded but notcompact, as the collection of all singleton points of the space is an open cover which admits no finite subcover.It is complete but not totally bounded.

26 CHAPTER 6. COMPACT SPACE

Characterization by continuous functions

Let X be a topological space and C(X) the ring of real continuous functions on X. For each p∈X, the evaluation map

evp : C(X) → R

given by evp(f)=f(p) is a ring homomorphism. The kernel of evp is a maximal ideal, since the residue field C(X)/kerevp is the field of real numbers, by the first isomorphism theorem. A topological spaceX is pseudocompact if and onlyif every maximal ideal in C(X) has residue field the real numbers. For completely regular spaces, this is equivalent toevery maximal ideal being the kernel of an evaluation homomorphism.[7] There are pseudocompact spaces that arenot compact, though.In general, for non-pseudocompact spaces there are always maximal ideals m in C(X) such that the residue fieldC(X)/m is a (non-archimedean) hyperreal field. The framework of non-standard analysis allows for the followingalternative characterization of compactness:[8] a topological space X is compact if and only if every point x of thenatural extension *X is infinitely close to a point x0 of X (more precisely, x is contained in the monad of x0).

Hyperreal definition

A space X is compact if its natural extension *X (for example, an ultrapower) has the property that every point of *Xis infinitely close to a suitable point ofX ⊂ ∗X . For example, an open real interval X=(0,1) is not compact becauseits hyperreal extension *(0,1) contains infinitesimals, which are infinitely close to 0, which is not a point of X.

6.3.3 Compactness of subspaces

A subset K of a topological space X is called compact if it is compact as a subspace. Explicitly, this means that forevery arbitrary collection

Uαα∈A

of open subsets of X such that

K ⊂∪α∈A

Uα,

there is a finite subset J of A such that

K ⊂∪i∈J

Ui.

6.4 Properties of compact spaces

6.4.1 Functions and compact spaces

A continuous image of a compact space is compact.[9] This implies the extreme value theorem: a continuous real-valued function on a nonempty compact space is bounded above and attains its supremum.[10] (Slightly more generally,this is true for an upper semicontinuous function.) As a sort of converse to the above statements, the pre-image of acompact space under a proper map is compact.

6.4.2 Compact spaces and set operations

A closed subset of a compact space is compact.,[11] and a finite union of compact sets is compact.

6.5. EXAMPLES 27

The product of any collection of compact spaces is compact. (Tychonoff’s theorem, which is equivalent to the axiomof choice)Every topological space X is an open dense subspace of a compact space having at most one point more than X, bythe Alexandroff one-point compactification. By the same construction, every locally compact Hausdorff space X isan open dense subspace of a compact Hausdorff space having at most one point more than X.

6.4.3 Ordered compact spaces

A nonempty compact subset of the real numbers has a greatest element and a least element.Let X be a simply ordered set endowed with the order topology. Then X is compact if and only if X is a completelattice (i.e. all subsets have suprema and infima).[12]

6.5 Examples• Any finite topological space, including the empty set, is compact. More generally, any space with a finitetopology (only finitely many open sets) is compact; this includes in particular the trivial topology.

• Any space carrying the cofinite topology is compact.

• Any locally compact Hausdorff space can be turned into a compact space by adding a single point to it, bymeans of Alexandroff one-point compactification. The one-point compactification of R is homeomorphic tothe circle S1; the one-point compactification of R2 is homeomorphic to the sphere S2. Using the one-pointcompactification, one can also easily construct compact spaces which are not Hausdorff, by starting with anon-Hausdorff space.

• The right order topology or left order topology on any bounded totally ordered set is compact. In particular,Sierpinski space is compact.

• R, carrying the lower limit topology, satisfies the property that no uncountable set is compact.

• In the cocountable topology on an uncountable set, no infinite set is compact. Like the previous example, thespace as a whole is not locally compact but is still Lindelöf.

• The closed unit interval [0,1] is compact. This follows from the Heine–Borel theorem. The open interval (0,1)is not compact: the open cover

(1

n, 1− 1

n

)for n = 3, 4, … does not have a finite subcover. Similarly, the set of rational numbers in the closedinterval [0,1] is not compact: the sets of rational numbers in the intervals[0,

1

π− 1

n

]and

[1

π+

1

n, 1

]cover all the rationals in [0, 1] for n = 4, 5, … but this cover does not have a finite subcover. (Note thatthe sets are open in the subspace topology even though they are not open as subsets of R.)

• The set R of all real numbers is not compact as there is a cover of open intervals that does not have a finitesubcover. For example, intervals (n−1, n+1) , where n takes all integer values in Z, cover R but there is nofinite subcover.

• For every natural number n, the n-sphere is compact. Again from the Heine–Borel theorem, the closed unitball of any finite-dimensional normed vector space is compact. This is not true for infinite dimensions; in fact,a normed vector space is finite-dimensional if and only if its closed unit ball is compact.

• On the other hand, the closed unit ball of the dual of a normed space is compact for the weak-* topology.(Alaoglu’s theorem)

28 CHAPTER 6. COMPACT SPACE

• The Cantor set is compact. In fact, every compact metric space is a continuous image of the Cantor set.

• Consider the set K of all functions f : R→ [0,1] from the real number line to the closed unit interval, and definea topology on K so that a sequence fn in K converges towards f ∈ K if and only if fn(x) convergestowards f(x) for all real numbers x. There is only one such topology; it is called the topology of pointwiseconvergence or the product topology. Then K is a compact topological space; this follows from the Tychonofftheorem.

• Consider the set K of all functions f : [0,1] → [0,1] satisfying the Lipschitz condition |f(x) − f(y)| ≤ |x − y| forall x, y ∈ [0,1]. Consider on K the metric induced by the uniform distance

d(f, g) = supx∈[0,1]

|f(x)− g(x)|.

Then by Arzelà–Ascoli theorem the space K is compact.

• The spectrum of any bounded linear operator on a Banach space is a nonempty compact subset of the complexnumbers C. Conversely, any compact subset of C arises in this manner, as the spectrum of some boundedlinear operator. For instance, a diagonal operator on the Hilbert space ℓ2 may have any compact nonemptysubset of C as spectrum.

6.5.1 Algebraic examples

• Compact groups such as an orthogonal group are compact, while groups such as a general linear group are not.

• Since the p-adic integers are homeomorphic to the Cantor set, they form a compact set.

• The spectrum of any commutative ring with the Zariski topology (that is, the set of all prime ideals) is compact,but never Hausdorff (except in trivial cases). In algebraic geometry, such topological spaces are examples ofquasi-compact schemes, “quasi” referring to the non-Hausdorff nature of the topology.

• The spectrum of a Boolean algebra is compact, a fact which is part of the Stone representation theorem. Stonespaces, compact totally disconnected Hausdorff spaces, form the abstract framework in which these spectraare studied. Such spaces are also useful in the study of profinite groups.

• The structure space of a commutative unital Banach algebra is a compact Hausdorff space.

• The Hilbert cube is compact, again a consequence of Tychonoff’s theorem.

• A profinite group (e.g., Galois group) is compact.

6.6 See also

• Compactly generated space

• Eberlein compactum

• Exhaustion by compact sets

• Lindelöf space

• Metacompact space

• Noetherian space

• Orthocompact space

• Paracompact space

6.7. NOTES 29

6.7 Notes[1] Kline 1972, pp. 952–953; Boyer & Merzbach 1991, p. 561

[2] Kline 1972, Chapter 46, §2

[3] Frechet, M. 1904. Generalisation d'un theorem de Weierstrass. Analyse Mathematique.

[4] (Kelley 1955, p. 163)

[5] Arkhangel’skii & Fedorchuk 1990, Theorem 5.3.7

[6] Willard 1970 Theorem 30.7.

[7] Gillman & Jerison 1976, §5.6

[8] Robinson, Theorem 4.1.13

[9] Arkhangel’skii &Fedorchuk 1990, Theorem 5.2.2; See also Compactness is preserved under a continuousmap at PlanetMath.org.

[10] Arkhangel’skii & Fedorchuk 1990, Corollary 5.2.1

[11] Arkhangel’skii & Fedorchuk 1990, Theorem 5.2.3; Closed set in a compact space is compact at PlanetMath.org. ; Closedsubsets of a compact set are compact at PlanetMath.org.

[12] (Steen & Seebach 1995, p. 67)

6.8 References• Alexandrov, Pavel; Urysohn, Pavel (1929), “Mémoire sur les espaces topologiques compacts”, KoninklijkeNederlandse Akademie van Wetenschappen te Amsterdam, Proceedings of the section of mathematical sciences14.

• Arkhangel’skii, A.V.; Fedorchuk, V.V. (1990), “The basic concepts and constructions of general topology”,in Arkhangel’skii, A.V.; Pontrjagin, L.S., General topology I, Encyclopedia of the Mathematical Sciences 17,Springer, ISBN 978-0-387-18178-3.

• Arkhangel’skii, A.V. (2001), “Compact space”, inHazewinkel, Michiel, Encyclopedia ofMathematics, Springer,ISBN 978-1-55608-010-4.

• Bolzano, Bernard (1817), Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwey Werthen, die ein ent-gegengesetzes Resultat gewähren, wenigstens eine reele Wurzel der Gleichung liege, Wilhelm Engelmann (Purelyanalytic proof of the theorem that between any two values which give results of opposite sign, there lies at leastone real root of the equation).

• Borel, Émile (1895), “Sur quelques points de la théorie des fonctions”, Annales Scientifiques de l'École NormaleSupérieure, 3 12: 9–55, JFM 26.0429.03

• Boyer, Carl B. (1959), The history of the calculus and its conceptual development, New York: Dover Publica-tions, MR 0124178.

• Arzelà, Cesare (1895), “Sulle funzioni di linee”, Mem. Accad. Sci. Ist. Bologna Cl. Sci. Fis. Mat. 5 (5):55–74.

• Arzelà, Cesare (1882–1883), “Un'osservazione intorno alle serie di funzioni”, Rend. Dell' Accad. R. Delle Sci.Dell'Istituto di Bologna: 142–159.

• Ascoli, G. (1883–1884), “Le curve limiti di una varietà data di curve”, Atti della R. Accad. Dei Lincei Memoriedella Cl. Sci. Fis. Mat. Nat. 18 (3): 521–586.

• Fréchet, Maurice (1906), “Sur quelques points du calcul fonctionnel”, Rendiconti del Circolo Matematico diPalermo 22 (1): 1–72, doi:10.1007/BF03018603.

• Gillman, Leonard; Jerison, Meyer (1976), Rings of continuous functions, Springer-Verlag.

• Kelley, John (1955), General topology, Graduate Texts in Mathematics 27, Springer-Verlag.

30 CHAPTER 6. COMPACT SPACE

• Kline, Morris (1972), Mathematical thought from ancient to modern times (3rd ed.), Oxford University Press(published 1990), ISBN 978-0-19-506136-9.

• Lebesgue, Henri (1904), Leçons sur l'intégration et la recherche des fonctions primitives, Gauthier-Villars.

• Robinson, Abraham (1996), Non-standard analysis, Princeton University Press, ISBN 978-0-691-04490-3,MR 0205854.

• Scarborough, C.T.; Stone, A.H. (1966), “Products of nearly compact spaces”, Transactions of the AmericanMathematical Society (Transactions of the American Mathematical Society, Vol. 124, No. 1) 124 (1): 131–147, doi:10.2307/1994440, JSTOR 1994440.

• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 507446

• Willard, Stephen (1970), General Topology, Dover publications, ISBN 0-486-43479-6

6.9 External links• Countably compact at PlanetMath.org.

• Sundström, Manya Raman (2010). “A pedagogical history of compactness”. v1. arXiv:1006.4131 [math.HO].

This article incorporates material from Examples of compact spaces on PlanetMath, which is licensed under the CreativeCommons Attribution/Share-Alike License.

Chapter 7

Compactly embedded

In mathematics, the notion of being compactly embedded expresses the idea that one set or space is “well contained”inside another. There are versions of this concept appropriate to general topology and functional analysis.

7.1 Definition (topological spaces)

Let (X, T) be a topological space, and let V and W be subsets of X. We say that V is compactly embedded in W,and write V ⊂⊂W, if

• V ⊆ Cl(V) ⊆ Int(W), where Cl(V) denotes the closure of V, and Int(W) denotes the interior ofW ; and

• Cl(V) is compact.

7.2 Definition (normed spaces)

Let X and Y be two normed vector spaces with norms ||•||X and ||•||Y respectively, and suppose that X ⊆ Y. We saythat X is compactly embedded in Y, and write X ⊂⊂ Y, if

• X is continuously embedded in Y; i.e., there is a constant C such that ||x||Y ≤ C||x||X for all x in X; and

• The embedding of X into Y is a compact operator: any bounded set in X is totally bounded in Y, i.e. everysequence in such a bounded set has a subsequence that is Cauchy in the norm ||•||Y .

If Y is a Banach space, an equivalent definition is that the embedding operator (the identity) i : X → Y is a compactoperator.When applied to functional analysis, this version of compact embedding is usually used with Banach spaces of func-tions. Several of the Sobolev embedding theorems are compact embedding theorems.

7.3 References• Adams, Robert A. (1975). Sobolev Spaces. Boston, MA: Academic Press. ISBN 978-0-12-044150-1..

• Evans, Lawrence C. (1998). Partial differential equations. Providence, RI: American Mathematical Society.ISBN 0-8218-0772-2..

• Renardy, M., & Rogers, R. C. (1992). An Introduction to Partial Differential Equations. Berlin: Springer-Verlag. ISBN 3-540-97952-2..

31

Chapter 8

Cover (topology)

In mathematics, a cover of a set X is a collection of sets whose union contains X as a subset. Formally, if

C = Uα : α ∈ A

is an indexed family of sets Uα , then C is a cover of X if

X ⊆∪α∈A

Uα.

8.1 Cover in topology

Covers are commonly used in the context of topology. If the set X is a topological space, then a cover C of X is acollection of subsets Uα of X whose union is the whole space X. In this case we say that C covers X, or that the setsUα cover X. Also, if Y is a subset of X, then a cover of Y is a collection of subsets of X whose union contains Y, i.e.,C is a cover of Y if

Y ⊆∪α∈A

Uα

Let C be a cover of a topological space X. A subcover of C is a subset of C that still covers X.We say that C is an open cover if each of its members is an open set (i.e. each Uα is contained in T, where T is thetopology on X).A cover of X is said to be locally finite if every point of X has a neighborhood which intersects only finitely many setsin the cover. Formally, C = Uα is locally finite if for any x ∈ X, there exists some neighborhood N(x) of x suchthat the set

α ∈ A : Uα ∩N(x) = ∅

is finite. A cover of X is said to be point finite if every point of X is contained in only finitely many sets in the cover.(locally finite implies point finite)

8.2 Refinement

A refinement of a cover C of a topological space X is a new cover D of X such that every set in D is contained insome set in C. Formally,

32

8.3. COMPACTNESS 33

D = Vβ∈B

is a refinement of

Uα∈A when ∀β ∃α Vβ ⊆ Uα

In other words, there is a refinement map ϕ : B → A satisfying Vβ ⊆ Uϕ(β) for every β ∈ B . This map is used,for instance, in the Čech cohomology of X.[1]

Every subcover is also a refinement, but the opposite is not always true. A subcover is made from the sets that are inthe cover, but omitting some of them; whereas a refinement is made from any sets that are subsets of the sets in thecover.The refinement relation is a preorder on the set of covers of X.Generally speaking, a refinement of a given structure is another that in some sense contains it. Examples are to befound when partitioning an interval (one refinement of a0 < a1 < ... < an being a0 < b0 < a1 < a2 < ... < an <b1 ), considering topologies (the standard topology in euclidean space being a refinement of the trivial topology).When subdividing simplicial complexes (the first barycentric subdivision of a simplicial complex is a refinement), thesituation is slightly different: every simplex in the finer complex is a face of some simplex in the coarser one, andboth have equal underlying polyhedra.Yet another notion of refinement is that of star refinement.

8.3 Compactness

The language of covers is often used to define several topological properties related to compactness. A topologicalspace X is said to be

• Compact, if every open cover has a finite subcover, (or equivalently that every open cover has a finite refinement);

• Lindelöf, if every open cover has a countable subcover, (or equivalently that every open cover has a countablerefinement);

• Metacompact, if every open cover has a point finite open refinement;

• Paracompact, if every open cover admits a locally finite open refinement.

For some more variations see the above articles.

8.4 Covering dimension

A topological space X is said to be of covering dimension n if every open cover of X has a point finite open refinementsuch that no point of X is included in more than n+1 sets in the refinement and if n is the minimum value for whichthis is true.[2] If no such minimal n exists, the space is said to be of infinite covering dimension.

8.5 See also

• Covering space

• Atlas (topology)

• Set cover problem

34 CHAPTER 8. COVER (TOPOLOGY)

8.6 Notes[1] Bott, Tu (1982). Differential Forms in Algebraic Topology. p. 111.

[2] Munkres, James (1999). Topology (2nd ed.). Prentice Hall. ISBN 0-13-181629-2.

8.7 References1. Introduction to Topology, Second Edition, Theodore W. Gamelin & Robert Everist Greene. Dover Publications

1999. ISBN 0-486-40680-6

2. General Topology, John L. Kelley. D. Van Nostrand Company, Inc. Princeton, NJ. 1955.

8.8 External links• Hazewinkel, Michiel, ed. (2001), “Covering (of a set)", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Chapter 9

Euclidean space

This article is about Euclidean spaces of all dimensions. For 3-dimensional Euclidean space, see 3-dimensionalspace.In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space

of Euclidean geometry, and certain other spaces. It is named after the Ancient Greek mathematician Euclid ofAlexandria.[1] The term “Euclidean” distinguishes these spaces from other types of spaces considered in moderngeometry. Euclidean spaces also generalize to higher dimensions.Classical Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates,while the other properties of these spaces were deduced as theorems. Geometric constructions are also used to definerational numbers. When algebra and mathematical analysis became developed enough, this relation reversed and nowit is more common to define Euclidean space using Cartesian coordinates and the ideas of analytic geometry. It meansthat points of the space are specified with collections of real numbers, and geometric shapes are defined as equationsand inequalities. This approach brings the tools of algebra and calculus to bear on questions of geometry and has theadvantage that it generalizes easily to Euclidean spaces of more than three dimensions.From the modern viewpoint, there is essentially only one Euclidean space of each dimension. With Cartesian coor-dinates it is modelled by the real coordinate space (Rn) of the same dimension. In one dimension, this is the realline; in two dimensions, it is the Cartesian plane; and in higher dimensions it is a coordinate space with three or morereal number coordinates. Mathematicians denote the n-dimensional Euclidean space by En if they wish to emphasizeits Euclidean nature, but Rn is used as well since the latter is assumed to have the standard Euclidean structure, andthese two structures are not always distinguished. Euclidean spaces have finite dimension.[2]

9.1 Intuitive overview

One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms ofdistance and angle. For example, there are two fundamental operations (referred to as motions) on the plane. One istranslation, which means a shifting of the plane so that every point is shifted in the same direction and by the samedistance. The other is rotation about a fixed point in the plane, in which every point in the plane turns about that fixedpoint through the same angle. One of the basic tenets of Euclidean geometry is that two figures (usually consideredas subsets) of the plane should be considered equivalent (congruent) if one can be transformed into the other by somesequence of translations, rotations and reflections (see below).In order to make all of this mathematically precise, the theory must clearly define the notions of distance, angle,translation, and rotation for a mathematically described space. Even when used in physical theories, Euclidean spaceis an abstraction detached from actual physical locations, specific reference frames, measurement instruments, and soon. A purely mathematical definition of Euclidean space also ignores questions of units of length and other physicaldimensions: the distance in a “mathematical” space is a number, not something expressed in inches or metres. Thestandard way to define such space, as carried out in the remainder of this article, is to define the Euclidean plane asa two-dimensional real vector space equipped with an inner product.[2] The reason for working with arbitrary vectorspaces instead of Rn is that it is often preferable to work in a coordinate-free manner (that is, without choosing apreferred basis). For then:

• the vectors in the vector space correspond to the points of the Euclidean plane,

35

36 CHAPTER 9. EUCLIDEAN SPACE

A sphere, the most perfect spatial shape according to Pythagoreans, also is an important concept in modern understanding of Eu-clidean spaces

• the addition operation in the vector space corresponds to translation, and

• the inner product implies notions of angle and distance, which can be used to define rotation.

Once the Euclidean plane has been described in this language, it is actually a simple matter to extend its concept toarbitrary dimensions. For the most part, the vocabulary, formulae, and calculations are not made any more difficultby the presence of more dimensions. (However, rotations are more subtle in high dimensions, and visualizing high-dimensional spaces remains difficult, even for experienced mathematicians.)A Euclidean space is not technically a vector space but rather an affine space, on which a vector space acts by transla-tions, or, conversely, a Euclidean vector is the difference (displacement) in an ordered pair of points, not a single point.Intuitively, the distinction says merely that there is no canonical choice of where the origin should go in the space,because it can be translated anywhere. When a certain point is chosen, it can be declared the origin and subsequentcalculations may ignore the difference between a point and its coordinate vector, as said above. See point–vectordistinction for details.

9.2. EUCLIDEAN STRUCTURE 37

X

Y

Z

O

xy

z

(x,y,z)

Every point in three-dimensional Euclidean space is determined by three coordinates.

9.2 Euclidean structure

These are distances between points and the angles between lines or vectors, which satisfy certain conditions (seebelow), which makes a set of points a Euclidean space. The natural way to obtain these quantities is by introducingand using the standard inner product (also known as the dot product) on Rn.[2] The inner product of any two realn-vectors x and y is defined by

x · y =n∑

i=1

xiyi = x1y1 + x2y2 + · · ·+ xnyn,

where xᵢ and yᵢ are ith coordinates of vectors x and y respectively. The result is always a real number.

9.2.1 Distance

Main article: Euclidean distance

The inner product of x with itself is always non-negative. This product allows us to define the “length” of a vector xthrough square root:

38 CHAPTER 9. EUCLIDEAN SPACE

∥x∥ =√x · x =

√√√√ n∑i=1

(xi)2.

This length function satisfies the required properties of a norm and is called the Euclidean norm on Rn.Finally, one can use the norm to define a metric (or distance function) on Rn by

d(x, y) = ∥x− y∥ =

√√√√ n∑i=1

(xi − yi)2.

This distance function is called the Euclidean metric. This formula expresses a special case of the Pythagoreantheorem.This distance function (which makes a metric space) is sufficient to define all Euclidean geometry, including the dotproduct. Thus, a real coordinate space together with this Euclidean structure is called Euclidean space. Its vectorsform an inner product space (in fact a Hilbert space), and a normed vector space.The metric space structure is the main reason behind the use of real numbers R, not some other ordered field, asthe mathematical foundation of Euclidean (and many other) spaces. Euclidean space is a complete metric space, aproperty which is impossible to achieve operating over rational numbers, for example.

9.2.2 Angle

Main article: AngleThe (non-reflex) angle θ (0° ≤ θ ≤ 180°) between vectors x and y is then given by

θ = arccos( x · y∥x∥∥y∥

)where arccos is the arccosine function. It is useful only for n > 1,[footnote 1] and the case n = 2 is somewhat special.Namely, on an oriented Euclidean plane one can define an angle between two vectors as a number defined modulo 1turn (usually denoted as either 2π or 360°), such that ∠y x = −∠x y. This oriented angle is equal either to the angleθ from the formula above or to −θ. If one non-zero vector is fixed (such as the first basis vector), then each non-zerovector is uniquely defined by its magnitude and angle.The angle does not change if vectors x and y are multiplied by positive numbers.Unlike the aforementioned situation with distance, the scale of angles is the same in pure mathematics, physics, andcomputing. It does not depend on the scale of distances; all distances may be multiplied by some fixed factor, andall angles will be preserved. Usually, the angle is considered a dimensionless quantity, but there are different unitsof measurement, such as radian (preferred in pure mathematics and theoretical physics) and degree (°) (preferred inmost applications).

9.2.3 Rotations and reflections

Main articles: Rotation (mathematics), Reflection (mathematics) and Orthogonal groupSee also: rotational symmetry and reflection symmetry

Symmetries of a Euclidean space are transformations which preserve the Euclidean metric (called isometries). Al-though aforementioned translations are most obvious of them, they have the same structure for any affine space anddo not show a distinctive character of Euclidean geometry. Another family of symmetries leave one point fixed,which may be seen as the origin without loss of generality. All transformations, which preserves the origin and theEuclidean metric, are linear maps. Such transformations Q must, for any x and y, satisfy:

9.2. EUCLIDEAN STRUCTURE 39

45°-315°405°

Positive and negative angles on the oriented plane

Qx ·Qy = x · y (explain the notation),|Qx| = |x|.

Such transforms constitute a group called the orthogonal group O(n). Its elements Q are exactly solutions of a matrixequation

QTQ = QQT = I,

where QT is the transpose of Q and I is the identity matrix.But a Euclidean space is orientable.[footnote 2] Each of these transformations either preserves or reverses orientationdepending on whether its determinant is +1 or −1 respectively. Only transformations which preserve orientation,which form the special orthogonal group SO(n), are considered (proper) rotations. This group has, as a Lie group,the same dimension n(n − 1) /2 and is the identity component of O(n).

40 CHAPTER 9. EUCLIDEAN SPACE

Groups SO(n) are well-studied for n ≤ 4. There are no non-trivial rotations in 0- and 1-spaces. Rotations of aEuclidean plane (n = 2) are parametrized by the angle (modulo 1 turn). Rotations of a 3-space are parametrized withaxis and angle, whereas a rotation of a 4-space is a superposition of two 2-dimensional rotations around perpendicularplanes.Among linear transforms in O(n) which reverse the orientation are hyperplane reflections. This is the only possiblecase for n ≤ 2, but starting from three dimensions, such isometry in the general position is a rotoreflection.

9.2.4 Euclidean group

Main article: Euclidean group

The Euclidean group E(n), also referred to as the group of all isometries ISO(n), treats translations, rotations, andreflections in a uniform way, considering them as group actions in the context of group theory, and especially in Liegroup theory. These group actions preserve the Euclidean structure.As the group of all isometries, ISO(n), the Euclidean group is important because it makes Euclidean geometry a caseof Klein geometry, a theoretical framework including many alternative geometries.The structure of Euclidean spaces – distances, lines, vectors, angles (up to sign), and so on – is invariant under thetransformations of their associated Euclidean group. For instance, translations form a commutative subgroup thatacts freely and transitively on En, while the stabilizer of any point there is the aforementioned O(n).Along with translations, rotations, reflections, as well as the identity transformation, Euclidean motions comprise alsoglide reflections (for n ≥ 2), screw operations and rotoreflections (for n ≥ 3), and even more complex combinationsof primitive transformations for n ≥ 4.The group structure determines which conditions a metric space needs to satisfy to be a Euclidean space:

1. Firstly, a metric space must be translationally invariant with respect to some (finite-dimensional) real vectorspace. This means that the space itself is an affine space, that the space is flat, not curved, and points do nothave different properties, and so any point can be translated to any other point.

2. Secondly, the metric must correspond in the aforementioned way to some positive-defined quadratic form onthis vector space, because point stabilizers have to be isomorphic to O(n).

9.3 Non-Cartesian coordinates

Main article: Coordinate system

Cartesian coordinates are arguably the standard, but not the only possible option for a Euclidean space. Skew coordi-nates are compatible with the affine structure of En, but make formulae for angles and distances more complicated.Another approach, which goes in line with ideas of differential geometry and conformal geometry, is orthogonalcoordinates, where coordinate hypersurfaces of different coordinates are orthogonal, although curved. Examplesinclude the polar coordinate system on Euclidean plane, the second important plane coordinate system.See below about expression of the Euclidean structure in curvilinear coordinates.

9.4 Geometric shapes

See also: List of mathematical shapes

9.4. GEOMETRIC SHAPES 41

3-dimensional skew coordinates

Parabolic coordinates

9.4.1 Lines, planes, and other subspaces

Main article: Flat (geometry)

The simplest (after points) objects in Euclidean space are flats, or Euclidean subspaces of lesser dimension. Pointsare 0-dimensional flats, 1-dimensional flats are called (straight) lines, and 2-dimensional flats are planes. (n − 1)-dimensional flats are called hyperplanes.

42 CHAPTER 9. EUCLIDEAN SPACE

Barycentric coordinates in 3-dimensional space: four coordinates are related with one linear equation

Any two distinct points lie on exactly one line. Any line and a point outside it lie on exactly one plane. Moregenerally, the properties of flats and their incidence of Euclidean space are shared with affine geometry, whereas theaffine geometry is devoid of distances and angles.

9.4.2 Line segments and triangles

Main articles: Line segment and Triangle geometry

This is not only a line which a pair (A, B) of distinct points defines. Points on the line which lie between A and B,together with A and B themselves, constitute a line segment A B. Any line segment has the length, which equals todistance between A and B. If A = B, then the segment is degenerate and its length equals to 0, otherwise the lengthis positive.A (non-degenerate) triangle is defined by three points not lying on the same line. Any triangle lies on one plane. Theconcept of triangle is not specific to Euclidean spaces, but Euclidean triangles have numerous special properties anddefine many derived objects.A triangle can be thought of as a 3-gon on a plane, a special (and the first meaningful in Euclidean geometry) case ofa polygon.

9.4. GEOMETRIC SHAPES 43

Three mutually transversal planes in the 3-dimensional space and their intersections, three lines

9.4.3 Polytopes and root systems

Main articles: Polytope and Root systemSee also: List of polygons, polyhedra and polytopes and List of regular polytopes

Polytope is a concept that generalizes polygons on a plane and polyhedra in 3-dimensional space (which are amongthe earliest studied geometrical objects). A simplex is a generalization of a line segment (1-simplex) and a triangle(2-simplex). A tetrahedron is a 3-simplex.The concept of a polytope belongs to affine geometry, which is more general than Euclidean. But Euclidean geometrydistinguish regular polytopes. For example, affine geometry does not see the difference between an equilateral triangleand a right triangle, but in Euclidean space the former is regular and the latter is not.Root systems are special sets of Euclidean vectors. A root system is often identical to the set of vertices of a regularpolytope.

9.4.4 Curves

Main article: Euclidean geometry of curvesSee also: List of curves

9.4.5 Balls, spheres, and hypersurfaces

Main articles: Ball (mathematics) and HypersurfaceSee also: n-sphere and List of surfaces

44 CHAPTER 9. EUCLIDEAN SPACE

9.5 Topology

Main article: Real coordinate space § Topological properties

Since Euclidean space is a metric space, it is also a topological space with the natural topology induced by the metric.The metric topology on En is called the Euclidean topology, and it is identical to the standard topology on Rn. Aset is open if and only if it contains an open ball around each of its points; in other words, open balls form a baseof the topology. The topological dimension of the Euclidean n-space equals n, which implies that spaces of differentdimension are not homeomorphic. A finer result is the invariance of domain, which proves that any subset of n-space,that is (with its subspace topology) homeomorphic to an open subset of n-space, is itself open.

9.6 Applications

Aside from countless uses in fundamental mathematics, a Euclidean model of the physical space can be used to solvemany practical problems with sufficient precision. Two usual approaches are a fixed, or stationary reference frame(i.e. the description of a motion of objects as their positions that change continuously with time), and the use ofGalilean space-time symmetry (such as in Newtonian mechanics). To both of them the modern Euclidean geometryprovides a convenient formalism; for example, the space of Galilean velocities is itself a Euclidean space (see relativevelocity for details).Topographical maps and technical drawings are planar Euclidean. An idea behind them is the scale invariance ofEuclidean geometry, that permits to represent large objects in a small sheet of paper, or a screen.

9.7 Alternatives and generalizations

Although Euclidean spaces are no longer considered to be the only possible setting for a geometry, they act as pro-totypes for other geometric objects. Ideas and terminology from Euclidean geometry (both traditional and analytic)are pervasive in modern mathematics, where other geometric objects share many similarities with Euclidean spaces,share part of their structure, or embed Euclidean spaces.

9.7.1 Curved spaces

Main article: Riemannian geometry

A smooth manifold is a Hausdorff topological space that is locally diffeomorphic to Euclidean space. Diffeomor-phism does not respect distance and angle, but if one additionally prescribes a smoothly varying inner product onthe manifold’s tangent spaces, then the result is what is called a Riemannian manifold. Put differently, a Riemannianmanifold is a space constructed by deforming and patching together Euclidean spaces. Such a space enjoys notionsof distance and angle, but they behave in a curved, non-Euclidean manner. The simplest Riemannian manifold, con-sisting of Rn with a constant inner product, is essentially identical to Euclidean n-space itself. Less trivial examplesare n-sphere and hyperbolic spaces. Discovery of the latter in the 19th century was branded as the non-Euclideangeometry.Also, the concept of a Riemannianmanifold permits an expression of the Euclidean structure in any smooth coordinatesystem, via metric tensor. From this tensor one can compute the Riemann curvature tensor. Where the latter equals tozero, the metric structure is locally Euclidean (it means that at least some open set in the coordinate space is isometricto a piece of Euclidean space), no matter whether coordinates are affine or curvilinear.

9.7.2 Indefinite quadratic form

See also: Sylvester’s law of inertia

9.8. SEE ALSO 45

If one replaces the inner product of a Euclidean space with an indefinite quadratic form, the result is a pseudo-Euclidean space. Smooth manifolds built from such spaces are called pseudo-Riemannian manifolds. Perhaps theirmost famous application is the theory of relativity, where flat spacetime is a pseudo-Euclidean space calledMinkowskispace, where rotations correspond to motions of hyperbolic spaces mentioned above. Further generalization to curvedspacetimes form pseudo-Riemannian manifolds, such as in general relativity.

9.7.3 Other number fields

Another line of generalization is to consider other number fields than one of real numbers. Over complex numbers, aHilbert space can be seen as a generalization of Euclidean dot product structure, although the definition of the innerproduct becomes a sesquilinear form for compatibility with metric structure.

9.7.4 Infinite dimensions

Main articles: inner product space and Hilbert space

9.8 See also• Function of several real variables, a coordinate presentation of a function on a Euclidean space

• Geometric algebra, an alternative algebraic formalism

• Vector calculus, a standard algebraic formalism

9.9 Footnotes[1] On the real line (n = 1) any two non-zero vectors are either parallel or antiparallel depending on whether their signs match

or oppose. There are no angles between 0 and 180°.

[2] It is Rn which is oriented because of the ordering of elements of the standard basis. Although an orientation is not anattribute of the Euclidean structure, there are only two possible orientations, and any linear automorphism either keepsorientation or reverses (swaps the two).

9.10 References[1] Ball, W.W. Rouse (1960) [1908]. A Short Account of the History of Mathematics (4th ed.). Dover Publications. pp. 50–62.

ISBN 0-486-20630-0.

[2] E.D. Solomentsev (7 February 2011). “Euclidean space.”. Encyclopedia of Mathematics. Springer. Retrieved 1 May 2014.

9.11 External links• Hazewinkel, Michiel, ed. (2001), “Euclidean space”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Chapter 10

Exhaustion by compact sets

In mathematics, especially analysis, exhaustion by compact sets of an open set E in the Euclidean space Rn (or amanifold with countable base) is an increasing sequence of compact sets Kj , where by increasing we meanKj is asubset ofKj+1 , with the limit (union) of the sequence being E.Sometimes one requires the sequence of compact sets to satisfy one more property— that Kj is contained in theinterior ofKj+1 for each j . This, however, is dispensed in Rn or a manifold with countable base.For example, consider a unit open disk and the concentric closed disk of each radius inside. That is letE = z; |z| <1 andKj = z; |z| ≤ (1− 1/j) . Then taking the limit (union) of the sequenceKj gives E. The example can beeasily generalized in other dimensions.

10.1 See also• σ-compact space

10.2 References• Leon Ehrenpreis, Theory of Distributions for Locally Compact Spaces, American Mathematical Society, 1982.ISBN 0-8218-1221-1.

10.3 External links• Exhaustion by compact sets at PlanetMath.org.

46

Chapter 11

Feebly compact space

In mathematics, a topological space is feebly compact if every locally finite cover by nonempty open sets is finite.Some facts:

• Every compact space is feebly compact.

• Every feebly compact paracompact space is compact.

• Every feebly compact space is pseudocompact but the converse is not necessarily true.

• For a completely regular Hausdorff space the properties of being feebly compact and pseudocompact are equiv-alent.

• Any maximal feebly compact space is submaximal.

47

Chapter 12

Functional analysis

For the assessment and treatment of human behavior, see Functional analysis (psychology).Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces

One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator ona function space, a common construction in functional analysis.

endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear operatorsacting upon these spaces and respecting these structures in a suitable sense. The historical roots of functional analysislie in the study of spaces of functions and the formulation of properties of transformations of functions such as theFourier transform as transformations defining continuous, unitary etc. operators between function spaces. This pointof view turned out to be particularly useful for the study of differential and integral equations.The usage of the word functional goes back to the calculus of variations, implying a function whose argument is afunction and the name was first used in Hadamard's 1910 book on that subject. However, the general concept ofa functional had previously been introduced in 1887 by the Italian mathematician and physicist Vito Volterra. Thetheory of nonlinear functionals was continued by students of Hadamard, in particular Fréchet and Lévy. Hadamardalso founded the modern school of linear functional analysis further developed by Riesz and the group of Polishmathematicians around Stefan Banach.In modern introductory texts to functional analysis, the subject is seen as the study of vector spaces endowed witha topology, in particular infinite dimensional spaces. In contrast, linear algebra deals mostly with finite dimensionalspaces, and does not use topology. An important part of functional analysis is the extension of the theory of measure,integration, and probability to infinite dimensional spaces, also known as infinite dimensional analysis.

48

12.1. NORMED VECTOR SPACES 49

12.1 Normed vector spaces

The basic and historically first class of spaces studied in functional analysis are complete normed vector spaces overthe real or complex numbers. Such spaces are called Banach spaces. An important example is a Hilbert space, wherethe norm arises from an inner product. These spaces are of fundamental importance in many areas, including themathematical formulation of quantum mechanics.More generally, functional analysis includes the study of Fréchet spaces and other topological vector spaces notendowed with a norm.An important object of study in functional analysis are the continuous linear operators defined on Banach and Hilbertspaces. These lead naturally to the definition of C*-algebras and other operator algebras.

12.1.1 Hilbert spaces

Hilbert spaces can be completely classified: there is a unique Hilbert space up to isomorphism for every cardinalityof the orthonormal basis. Finite-dimensional Hilbert spaces are fully understood in linear algebra, and infinite-dimensional separable Hilbert spaces are isomorphic to ℓ 2(ℵ0) . Separability being important for applications,functional analysis of Hilbert spaces consequently mostly deals with this space. One of the open problems in func-tional analysis is to prove that every bounded linear operator on a Hilbert space has a proper invariant subspace. Manyspecial cases of this invariant subspace problem have already been proven.

12.1.2 Banach spaces

General Banach spaces are more complicated than Hilbert spaces, and cannot be classified in such a simple manneras those. In particular, many Banach spaces lack a notion analogous to an orthonormal basis.Examples of Banach spaces are L p -spaces for any real number p ≥ 1 . Given also a measure µ on set X , thenL p(X) , sometimes also denoted L p(X,µ) or L p(µ) , has as its vectors equivalence classes [ f ] of measurablefunctions whose absolute value's p -th power has finite integral, that is, functions f for which one has

∫X

|f(x)|p dµ(x) < +∞

If µ is the counting measure, then the integral may be replaced by a sum. That is, we require

∑x∈X

|f(x)|p < +∞

Then it is not necessary to deal with equivalence classes, and the space is denoted ℓ p(X) , written more simply ℓ pin the case when X is the set of non-negative integers.In Banach spaces, a large part of the study involves the dual space: the space of all continuous linear maps from thespace into its underlying field, so-called functionals. A Banach space can be canonically identified with a subspace ofits bidual, which is the dual of its dual space. The corresponding map is an isometry but in general not onto. A generalBanach space and its bidual need not even be isometrically isomorphic in any way, contrary to the finite-dimensionalsituation. This is explained in the dual space article.Also, the notion of derivative can be extended to arbitrary functions between Banach spaces. See, for instance, theFréchet derivative article.

12.2 Major and foundational results

Important results of functional analysis include:

50 CHAPTER 12. FUNCTIONAL ANALYSIS

12.2.1 Uniform boundedness principle

Main article: Banach-Steinhaus theorem

The uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functionalanalysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cor-nerstones of the field. In its basic form, it asserts that for a family of continuous linear operators (and thus boundedoperators) whose domain is a Banach space, pointwise boundedness is equivalent to uniform boundedness in operatornorm.The theorem was first published in 1927 by Stefan Banach and Hugo Steinhaus but it was also proven independentlyby Hans Hahn.

Theorem (Uniform Boundedness Principle). Let X be a Banach space and Y be a normed vectorspace. Suppose that F is a collection of continuous linear operators from X to Y. If for all x in X one has

supT∈F ∥T (x)∥Y <∞,

then

supT∈F ∥T∥B(X,Y ) <∞.

12.2.2 Spectral theorem

Main article: Spectral theorem

There are many theorems known as the spectral theorem, but one in particular has many applications in functionalanalysis. Let A be the operator of multiplication by t on L2[0, 1], that is

[Aφ](t) = tφ(t).

Theorem:[1] Let A be a bounded self-adjoint operator on a Hilbert space H. Then there is a measure space (X, Σ, μ)and a real-valued essentially bounded measurable function f on X and a unitary operator U:H → L2μ(X) such that

U∗TU = A

where T is the multiplication operator:

[Tφ](x) = f(x)φ(x).

and ∥T∥ = ∥f∥∞This is the beginning of the vast research area of functional analysis called operator theory; see also the spectralmeasure.There is also an analogous spectral theorem for bounded normal operators on Hilbert spaces. The only difference inthe conclusion is that now f may be complex-valued.

12.2.3 Hahn-Banach theorem

Main article: Hahn-Banach theorem

TheHahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionalsdefined on a subspace of some vector space to the whole space, and it also shows that there are “enough” continuouslinear functionals defined on every normed vector space to make the study of the dual space “interesting”.

12.3. FOUNDATIONS OF MATHEMATICS CONSIDERATIONS 51

Hahn–Banach Theorem:[2] If p : V → R is a sublinear function, and φ : U → R is a linear functional on a linearsubspace U ⊆ V which is dominated by p on U, i.e.

φ(x) ≤ p(x) ∀x ∈ U

then there exists a linear extension ψ : V → R of φ to the whole space V, i.e., there exists a linear functional ψ suchthat

ψ(x) = φ(x) ∀x ∈ U,

ψ(x) ≤ p(x) ∀x ∈ V.

12.2.4 Open mapping theorem

Main article: Open mapping theorem (functional analysis)

The open mapping theorem, also known as the Banach–Schauder theorem (named after Stefan Banach and JuliuszSchauder), is a fundamental result which states that if a continuous linear operator between Banach spaces is surjectivethen it is an open map. More precisely,:[2]

Open Mapping Theorem. If X and Y are Banach spaces and A : X → Y is a surjective continuouslinear operator, then A is an open map (i.e. if U is an open set in X, then A(U) is open in Y).

The proof uses the Baire category theorem, and completeness of both X and Y is essential to the theorem. Thestatement of the theorem is no longer true if either space is just assumed to be a normed space, but is true if X andY are taken to be Fréchet spaces.

12.2.5 Closed graph theorem

Main article: Closed graph theorem

The closed graph theorem states the following: If X is a topological space and Y is a compact Hausdorff space, thenthe graph of T is closed if and only if T is continuous.[3]

12.2.6 Other topics

List of functional analysis topics.

12.3 Foundations of mathematics considerations

Most spaces considered in functional analysis have infinite dimension. To show the existence of a vector space basisfor such spaces may require Zorn’s lemma. However, a somewhat different concept, Schauder basis, is usually morerelevant in functional analysis. Many very important theorems require the Hahn–Banach theorem, usually provedusing axiom of choice, although the strictly weaker Boolean prime ideal theorem suffices. The Baire category theorem,needed to prove many important theorems, also requires a form of axiom of choice.

12.4 Points of view

Functional analysis in its present form includes the following tendencies:

52 CHAPTER 12. FUNCTIONAL ANALYSIS

• Abstract analysis. An approach to analysis based on topological groups, topological rings, and topologicalvector spaces.

• Geometry of Banach spaces contains many topics. One is combinatorial approach connected with Jean Bour-gain; another is a characterization of Banach spaces in which various forms of the law of large numbers hold.

• Noncommutative geometry. Developed by Alain Connes, partly building on earlier notions, such as GeorgeMackey's approach to ergodic theory.

• Connection with quantummechanics. Either narrowly defined as inmathematical physics, or broadly interpretedby, e.g. Israel Gelfand, to include most types of representation theory.

12.5 See also

• List of functional analysis topics

• Spectral theory

12.6 References[1] Hall, B.C. (2013), Quantum Theory for Mathematicians, Springer, p. 147

[2] Rudin, Walter (1991). Functional analysis. McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5.

[3] Munkres, James (2000), Topology (2nd ed.), Upper Saddle River: Prentice Hall, pp. 163–172, ISBN 0-13-181629-2, p.171

12.7 Further reading

• Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis: A Hitchhiker’s Guide, 3rd ed., Springer 2007,ISBN 978-3-540-32696-0. Online doi:10.1007/3-540-29587-9 (by subscription)

• Bachman, G., Narici, L.: Functional analysis, Academic Press, 1966. (reprint Dover Publications)

• Banach S. Theory of Linear Operations. Volume 38, North-Holland Mathematical Library, 1987, ISBN 0-444-70184-2

• Brezis, H.: Analyse Fonctionnelle, Dunod ISBN 978-2-10-004314-9 or ISBN 978-2-10-049336-4

• Conway, J. B.: A Course in Functional Analysis, 2nd edition, Springer-Verlag, 1994, ISBN 0-387-97245-5

• Dunford, N. and Schwartz, J.T.: Linear Operators, General Theory, John Wiley & Sons, and other 3 volumes,includes visualization charts

• Edwards, R. E.: Functional Analysis, Theory and Applications, Hold, Rinehart and Winston, 1965.

• Eidelman, Yuli, Vitali Milman, and Antonis Tsolomitis: Functional Analysis: An Introduction, AmericanMathematical Society, 2004.

• Freidman, A.: Foundations of Modern Analysis, Dover Publications, Paperback Edition, July 21, 2010

• Giles,J.R.: Introduction to the Analysis of Normed Linear Spaces,Cambridge University Press,2000

• Hirsch F., Lacombe G. - “Elements of Functional Analysis”, Springer 1999.

• Hutson, V., Pym, J.S., Cloud M.J.: Applications of Functional Analysis and Operator Theory, 2nd edition,Elsevier Science, 2005, ISBN 0-444-51790-1

• Kantorovitz, S.,Introduction to Modern Analysis, Oxford University Press,2003,2nd ed.2006.

12.8. EXTERNAL LINKS 53

• Kolmogorov, A.N and Fomin, S.V.: Elements of the Theory of Functions and Functional Analysis, DoverPublications, 1999

• Kreyszig, E.: Introductory Functional Analysis with Applications, Wiley, 1989.

• Lax, P.: Functional Analysis, Wiley-Interscience, 2002, ISBN 0-471-55604-1

• Lebedev, L.P. and Vorovich, I.I.: Functional Analysis in Mechanics, Springer-Verlag, 2002

• Michel, Anthony N. and Charles J. Herget: Applied Algebra and Functional Analysis, Dover, 1993.

• Pietsch, Albrecht: History of Banach spaces and linear operators, Birkhauser Boston Inc., 2007, ISBN 978-0-8176-4367-6

• Reed, M., Simon, B.: “Functional Analysis”, Academic Press 1980.

• Riesz, F. and Sz.-Nagy, B.: Functional Analysis, Dover Publications, 1990

• Rudin, W.: Functional Analysis, McGraw-Hill Science, 1991

• Schechter, M.: Principles of Functional Analysis, AMS, 2nd edition, 2001

• Shilov, Georgi E.: Elementary Functional Analysis, Dover, 1996.

• Sobolev, S.L.: Applications of Functional Analysis in Mathematical Physics, AMS, 1963

• Yosida, K.: Functional Analysis, Springer-Verlag, 6th edition, 1980

• Vogt, D., Meise, R.: Introduction to Functional Analysis, Oxford University Press, 1997.

12.8 External links• Hazewinkel, Michiel, ed. (2001), “Functional analysis”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Topics in Real and Functional Analysis by Gerald Teschl, University of Vienna.

• Lecture Notes on Functional Analysis by Yevgeny Vilensky, New York University.

• Earliest Known Uses of Some of the Words of Mathematics: Calculus & Analysis by John Aldrich Universityof Southampton.

• Lecture videos on functional analysis by Greg Morrow from University of Colorado Colorado Springs

• An Introduction to Functional Analysis on Coursera by John Cagnol from Ecole Centrale Paris

Chapter 13

H-closed space

In mathematics, a topological space X is said to be H-closed, or Hausdorff closed, or absolutely closed if it isclosed in every Hausdorff space containing it as a subspace. This property is a generalization of compactness, sincea compact subset of a Hausdorff space is closed. Thus, every compact Hausdorff space is H-closed. The notion ofan H-closed space has been introduced in 1924 by P. Alexandroff and P. Urysohn.

13.1 Examples and equivalent formulations• The unit interval [0, 1] , endowed with the smallest topology which refines the euclidean topology, and containsQ ∩ [0, 1] as an open set is H-closed but not compact.

• Every regular Hausdorff H-closed space is compact.

• A Hausdorff space is H-closed if and only if every open cover has a finite subfamily with dense union.

13.2 See also• Compact space

13.3 References• K.P. Hart, Jun-iti Nagata, J.E. Vaughan (editors), Encyclopedia of General Topology, Chapter d20 (by JackPorter and Johannes Vermeer)

54

Chapter 14

Hasse diagram

x, y

x y

The power set of a 2-element set ordered by inclusion

In order theory, a Hasse diagram (/ˈhæsə/; German: /ˈhasə/) is a type of mathematical diagram used to represent afinite partially ordered set, in the form of a drawing of its transitive reduction. Concretely, for a partially ordered set(S, ≤) one represents each element of S as a vertex in the plane and draws a line segment or curve that goes upwardfrom x to y whenever y covers x (that is, whenever x < y and there is no z such that x < z < y). These curves maycross each other but must not touch any vertices other than their endpoints. Such a diagram, with labeled vertices,uniquely determines its partial order.Hasse diagrams are named after Helmut Hasse (1898–1979); according to Birkhoff (1948), they are so called becauseof the effective use Hasse made of them. However, Hasse was not the first to use these diagrams; they appear, e.g.,in Vogt (1895). Although Hasse diagrams were originally devised as a technique for making drawings of partiallyordered sets by hand, they have more recently been created automatically using graph drawing techniques.[1]

55

56 CHAPTER 14. HASSE DIAGRAM

The phrase “Hasse diagram” may also refer to the transitive reduction as an abstract directed acyclic graph, indepen-dently of any drawing of that graph, but this usage is eschewed here.

14.1 A “good” Hasse diagram

Although Hasse diagrams are simple as well as intuitive tools for dealing with finite posets, it turns out to be ratherdifficult to draw “good” diagrams. The reason is that there will in general be many possible ways to draw a Hassediagram for a given poset. The simple technique of just starting with the minimal elements of an order and thendrawing greater elements incrementally often produces quite poor results: symmetries and internal structure of theorder are easily lost.The following example demonstrates the issue. Consider the power set of a 4-element set ordered by inclusion ⊆ .Below are four different Hasse diagrams for this partial order. Each subset has a node labelled with a binary encodingthat shows whether a certain element is in the subset (1) or not (0):The first diagram makes clear that the power set is a graded poset. The second diagram has the same graded struc-ture, but by making some edges longer than others, it emphasizes that the 4-dimensional cube is a union of two3-dimensional cubes. The third diagram shows some of the internal symmetry of the structure. In the fourth diagramthe vertices are arranged like the elements of a 4×4 matrix.

14.2 Upward planarity

Main article: Upward planar drawingIf a partial order can be drawn as a Hasse diagram in which no two edges cross, its covering graph is said to be upwardplanar. A number of results on upward planarity and on crossing-free Hasse diagram construction are known:

• If the partial order to be drawn is a lattice, then it can be drawn without crossings if and only if it has orderdimension at most two.[2] In this case, a non-crossing drawing may be found by deriving Cartesian coordinatesfor the elements from their positions in the two linear orders realizing the order dimension, and then rotatingthe drawing counterclockwise by a 45-degree angle.

• If the partial order has at most one minimal element, or it has at most one maximal element, then it may betested in linear time whether it has a non-crossing Hasse diagram.[3]

• It is NP-complete to determine whether a partial order with multiple sources and sinks can be drawn as acrossing-free Hasse diagram.[4] However, finding a crossing-free Hasse diagram is fixed-parameter tractablewhen parametrized by the number of articulation points and triconnected components of the transitive reductionof the partial order.[5]

• If the y-coordinates of the elements of a partial order are specified, then a crossing-free Hasse diagram re-specting those coordinate assignments can be found in linear time, if such a diagram exists.[6] In particular,if the input poset is a graded poset, it is possible to determine in linear time whether there is a crossing-freeHasse diagram in which the height of each vertex is proportional to its rank.

14.3 Notes[1] E.g., see Di Battista & Tamassia (1988) and Freese (2004).

[2] Garg & Tamassia (1995a), Theorem 9, p. 118; Baker, Fishburn & Roberts (1971), theorem 4.1, page 18.

[3] Garg & Tamassia (1995a), Theorem 15, p. 125; Bertolazzi et al. (1993).

[4] Garg & Tamassia (1995a), Corollary 1, p. 132; Garg & Tamassia (1995b).

[5] Chan (2004).

[6] Jünger & Leipert (1999).

14.4. REFERENCES 57

This Hasse diagram of the lattice of subgroups of the dihedral group Dih4 has no crossing edges.

14.4 References• Baker, K. A.; Fishburn, P.; Roberts, F. S. (1971), “Partial orders of dimension 2”, Networks 2 (1): 11–28,doi:10.1002/net.3230020103.

• Bertolazzi, R; Di Battista, G.; Mannino, C.; Tamassia, R. (1993), “Optimal upward planarity testing of single-source digraphs”, Proc. 1st European Symposium on Algorithms (ESA '93), Lecture Notes in Computer Science726, Springer-Verlag, pp. 37–48, doi:10.1007/3-540-57273-2_42.

• Birkhoff, Garrett (1948), Lattice Theory (Revised ed.), American Mathematical Society.

• Chan, Hubert (2004), “A parameterized algorithm for upward planarity testing”, Proc. 12th European Sympo-sium on Algorithms (ESA '04), Lecture Notes in Computer Science 3221, Springer-Verlag, pp. 157–168.

• Di Battista, G.; Tamassia, R. (1988), “Algorithms for plane representation of acyclic digraphs”, TheoreticalComputer Science 61 (2–3): 175–178, doi:10.1016/0304-3975(88)90123-5.

• Freese, Ralph (2004), “Automated lattice drawing”, Concept Lattices, Lecture Notes in Computer Science2961, Springer-Verlag, pp. 589–590. An extended preprint is available online: .

• Garg, Ashim; Tamassia, Roberto (1995a), “Upward planarity testing”,Order 12 (2): 109–133, doi:10.1007/BF01108622.

• Garg, Ashim; Tamassia, Roberto (1995b), “On the computational complexity of upward and rectilinear pla-narity testing”, Graph Drawing (Proc. GD '94), LectureNotes in Computer Science 894, Springer-Verlag, pp.286–297, doi:10.1007/3-540-58950-3_384.

58 CHAPTER 14. HASSE DIAGRAM

• Jünger, Michael; Leipert, Sebastian (1999), “Level planar embedding in linear time”, Graph Drawing (Proc.GD '99), Lecture Notes in Computer Science 1731, pp. 72–81, doi:10.1007/3-540-46648-7_7, ISBN 978-3-540-66904-3.

• Vogt, Henri Gustav (1895), Leçons sur la résolution algébrique des équations, Nony, p. 91.

14.5 External links• Related media at Wikimedia Commons:

• Hasse diagram (Gallery)• Hasse diagrams (Category)

• Weisstein, Eric W., “Hasse Diagram”, MathWorld.

Chapter 15

Hausdorff space

In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topologicalspace in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on atopological space, the “Hausdorff condition” (T2) is the most frequently used and discussed. It implies the uniquenessof limits of sequences, nets, and filters.Hausdorff spaces are named after Felix Hausdorff, one of the founders of topology. Hausdorff’s original definitionof a topological space (in 1914) included the Hausdorff condition as an axiom.

15.1 Definitions

U

x

V

y

The points x and y, separated by their respective neighbourhoods U and V.

Points x and y in a topological space X can be separated by neighbourhoods if there exists a neighbourhood U of xand a neighbourhood V of y such that U and V are disjoint (U ∩ V = ∅). X is a Hausdorff space if any two distinctpoints of X can be separated by neighborhoods. This condition is the third separation axiom (after T0 and T1), whichis why Hausdorff spaces are also called T2 spaces. The name separated space is also used.A related, but weaker, notion is that of a preregular space. X is a preregular space if any two topologically distin-guishable points can be separated by neighbourhoods. Preregular spaces are also called R1 spaces.The relationship between these two conditions is as follows. A topological space is Hausdorff if and only if it is bothpreregular (i.e. topologically distinguishable points are separated by neighbourhoods) and Kolmogorov (i.e. distinctpoints are topologically distinguishable). A topological space is preregular if and only if its Kolmogorov quotient is

59

60 CHAPTER 15. HAUSDORFF SPACE

Hausdorff.

15.2 Equivalences

For a topological space X, the following are equivalent:

• X is a Hausdorff space.

• Limits of nets in X are unique.[1]

• Limits of filters on X are unique.[2]

• Any singleton set x ⊂ X is equal to the intersection of all closed neighbourhoods of x.[3] (A closed neigh-bourhood of x is a closed set that contains an open set containing x.)

• The diagonal Δ = (x,x) | x ∈ X is closed as a subset of the product space X × X.

15.3 Examples and counterexamples

Almost all spaces encountered in analysis are Hausdorff; most importantly, the real numbers (under the standardmetric topology on real numbers) are a Hausdorff space. More generally, all metric spaces are Hausdorff. In fact,many spaces of use in analysis, such as topological groups and topological manifolds, have the Hausdorff conditionexplicitly stated in their definitions.A simple example of a topology that is T1 but is not Hausdorff is the cofinite topology defined on an infinite set.Pseudometric spaces typically are not Hausdorff, but they are preregular, and their use in analysis is usually only in theconstruction of Hausdorff gauge spaces. Indeed, when analysts run across a non-Hausdorff space, it is still probablyat least preregular, and then they simply replace it with its Kolmogorov quotient, which is Hausdorff.In contrast, non-preregular spaces are encountered much more frequently in abstract algebra and algebraic geometry,in particular as the Zariski topology on an algebraic variety or the spectrum of a ring. They also arise in the modeltheory of intuitionistic logic: every complete Heyting algebra is the algebra of open sets of some topological space,but this space need not be preregular, much less Hausdorff.While the existence of unique limits for convergent nets and filters implies that a space is Hausdorff, there are non-Hausdorff T1 spaces in which every convergent sequence has a unique limit.[4]

15.4 Properties

Subspaces and products of Hausdorff spaces are Hausdorff,[5] but quotient spaces of Hausdorff spaces need not beHausdorff. In fact, every topological space can be realized as the quotient of some Hausdorff space.[6]

Hausdorff spaces are T1, meaning that all singletons are closed. Similarly, preregular spaces are R0.Another nice property of Hausdorff spaces is that compact sets are always closed.[7] This may fail in non-Hausdorffspaces such as Sierpiński space.The definition of a Hausdorff space says that points can be separated by neighborhoods. It turns out that this impliessomething which is seemingly stronger: in a Hausdorff space every pair of disjoint compact sets can also be separatedby neighborhoods,[8] in other words there is a neighborhood of one set and a neighborhood of the other, such that thetwo neighborhoods are disjoint. This is an example of the general rule that compact sets often behave like points.Compactness conditions together with preregularity often imply stronger separation axioms. For example, any locallycompact preregular space is completely regular. Compact preregular spaces are normal, meaning that they satisfyUrysohn’s lemma and the Tietze extension theorem and have partitions of unity subordinate to locally finite opencovers. The Hausdorff versions of these statements are: every locally compact Hausdorff space is Tychonoff, andevery compact Hausdorff space is normal Hausdorff.The following results are some technical properties regarding maps (continuous and otherwise) to and fromHausdorffspaces.

15.5. PREREGULARITY VERSUS REGULARITY 61

Let f : X → Y be a continuous function and suppose Y is Hausdorff. Then the graph of f, (x, f(x)) | x ∈ X , isa closed subset of X × Y.Let f : X → Y be a function and let ker(f) ≜ (x, x′) | f(x) = f(x′) be its kernel regarded as a subspace of X ×X.

• If f is continuous and Y is Hausdorff then ker(f) is closed.

• If f is an open surjection and ker(f) is closed then Y is Hausdorff.

• If f is a continuous, open surjection (i.e. an open quotient map) then Y is Hausdorff if and only if ker(f) isclosed.

If f,g : X→ Y are continuous maps and Y is Hausdorff then the equalizer eq(f, g) = x | f(x) = g(x) is closed inX. It follows that if Y is Hausdorff and f and g agree on a dense subset of X then f = g. In other words, continuousfunctions into Hausdorff spaces are determined by their values on dense subsets.Let f : X→ Y be a closed surjection such that f−1(y) is compact for all y ∈ Y. Then if X is Hausdorff so is Y.Let f : X→ Y be a quotient map with X a compact Hausdorff space. Then the following are equivalent

• Y is Hausdorff

• f is a closed map

• ker(f) is closed

15.5 Preregularity versus regularity

All regular spaces are preregular, as are all Hausdorff spaces. There are many results for topological spaces that holdfor both regular and Hausdorff spaces. Most of the time, these results hold for all preregular spaces; they were listedfor regular and Hausdorff spaces separately because the idea of preregular spaces came later. On the other hand,those results that are truly about regularity generally don't also apply to nonregular Hausdorff spaces.There are many situations where another condition of topological spaces (such as paracompactness or local com-pactness) will imply regularity if preregularity is satisfied. Such conditions often come in two versions: a regularversion and a Hausdorff version. Although Hausdorff spaces aren't generally regular, a Hausdorff space that is also(say) locally compact will be regular, because any Hausdorff space is preregular. Thus from a certain point of view,it is really preregularity, rather than regularity, that matters in these situations. However, definitions are usually stillphrased in terms of regularity, since this condition is better known than preregularity.See History of the separation axioms for more on this issue.

15.6 Variants

The terms “Hausdorff”, “separated”, and “preregular” can also be applied to such variants on topological spacesas uniform spaces, Cauchy spaces, and convergence spaces. The characteristic that unites the concept in all of theseexamples is that limits of nets and filters (when they exist) are unique (for separated spaces) or unique up to topologicalindistinguishability (for preregular spaces).As it turns out, uniform spaces, and more generally Cauchy spaces, are always preregular, so the Hausdorff condi-tion in these cases reduces to the T0 condition. These are also the spaces in which completeness makes sense, andHausdorffness is a natural companion to completeness in these cases. Specifically, a space is complete if and only ifevery Cauchy net has at least one limit, while a space is Hausdorff if and only if every Cauchy net has at most onelimit (since only Cauchy nets can have limits in the first place).

62 CHAPTER 15. HAUSDORFF SPACE

15.7 Algebra of functions

The algebra of continuous (real or complex) functions on a compact Hausdorff space is a commutative C*-algebra,and conversely by the Banach–Stone theorem one can recover the topology of the space from the algebraic propertiesof its algebra of continuous functions. This leads to noncommutative geometry, where one considers noncommutativeC*-algebras as representing algebras of functions on a noncommutative space.

15.8 Academic humour• Hausdorff condition is illustrated by the pun that in Hausdorff spaces any two points can be “housed off” fromeach other by open sets.[9]

• In the Mathematics Institute of at the University of Bonn, in which Felix Hausdorff researched and lectured,there is a certain room designated the Hausdorff-Raum. This is a pun, as Raum means both room and spacein German.

15.9 See also• Quasitopological space

• Weak Hausdorff space

• Fixed-point space, a Hausdorff space X such that every continuous function f:X→X has a fixed point.

15.10 Notes[1] Willard, pp. 86–87.

[2] Willard, pp. 86–87.

[3] Bourbaki, p. 75.

[4] van Douwen, Eric K. (1993). “An anti-Hausdorff Fréchet space in which convergent sequences have unique limits”.Topology and its Applications 51 (2): 147–158. doi:10.1016/0166-8641(93)90147-6.

[5] Hausdorff property is hereditary at PlanetMath.org.

[6] Shimrat, M. (1956). “Decomposition spaces and separation properties”. Quart. J. Math. 2: 128–129.

[7] Proof of A compact set in a Hausdorff space is closed at PlanetMath.org.

[8] Willard, p. 124.

[9] Colin Adams and Robert Franzosa. Introduction to Topology: Pure and Applied. p. 42

15.11 References• Arkhangelskii, A.V., L.S. Pontryagin, General Topology I, (1990) Springer-Verlag, Berlin. ISBN 3-540-18178-4.

• Bourbaki; Elements of Mathematics: General Topology, Addison-Wesley (1966).

• Hazewinkel, Michiel, ed. (2001), “Hausdorff space”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.

Chapter 16

Hemicompact space

In mathematics, in the field of topology, a topological space is said to be hemicompact if it has a sequence of compactsubsets such that every compact subset of the space lies inside some compact set in the sequence. Clearly, this forcesthe union of the sequence to be the whole space, because every point is compact and hence must lie in one of thecompact sets.

16.1 Examples• Every compact space is hemicompact.

• The real line is hemicompact.

• Every locally compact Lindelöf space is hemicompact.

16.2 Properties

Every hemicompact space is σ-compact and if in addition it is first countable then it is locally compact.If X is a hemicompact space, then the space C(X,M) of all continuous functions f : X → M to a metric space(M, δ) with the compact-open topology is metrizable. To see this, take a sequence K1,K2, . . . of compact subsetsof X such that every compact subset of X lies inside some compact set in this sequence (the existence of such asequence follows from the hemicompactness ofX ). Denote

dn(f, g) = supx∈Kn

δ(f(x), g(x))

for f, g ∈ C(X,M) and n ∈ N . Then

d(f, g) =∞∑

n=1

1

2n· dn(f, g)

1 + dn(f, g)

defines a metric on C(X,M) which induces the compact-open topology.

16.3 See also• Compact space

• Locally compact space

• Lindelöf space

63

64 CHAPTER 16. HEMICOMPACT SPACE

16.4 References• Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.

Chapter 17

Interior (topology)

S

x

y

The point x is an interior point of S. The point y is on the boundary of S.

In mathematics, specifically in topology, the interior of a subset S of points of a topological space X consists of allpoints of S that do not belong to the boundary of S. A point that is in the interior of S is an interior point of S.The interior of S is the complement of the closure of the complement of S. In this sense interior and closure are dual

65

66 CHAPTER 17. INTERIOR (TOPOLOGY)

notions.The exterior of a set is the interior of its complement, equivalently the complement of its closure; it consists of thepoints that are in neither the set nor its boundary. The interior, boundary, and exterior of a subset together partitionthe whole space into three blocks (or fewer when one or more of these is empty). The interior and exterior are alwaysopen while the boundary is always closed. Sets with empty interior have been called boundary sets.[1]

17.1 Definitions

17.1.1 Interior point

If S is a subset of a Euclidean space, then x is an interior point of S if there exists an open ball centered at x which iscompletely contained in S. (This is illustrated in the introductory section to this article.)This definition generalizes to any subset S of a metric space X with metric d: x is an interior point of S if there existsr > 0, such that y is in S whenever the distance d(x, y) < r.This definition generalises to topological spaces by replacing “open ball” with "open set". Let S be a subset of atopological space X. Then x is an interior point of S if x is contained in an open subset of S. (Equivalently, x is aninterior point of S if there exists a neighbourhood of x which is contained in S.)

17.1.2 Interior of a set

The interior of a set S is the set of all interior points of S. The interior of S is denoted int(S), Int(S), or So. Theinterior of a set has the following properties.

• int(S) is an open subset of S.

• int(S) is the union of all open sets contained in S.

• int(S) is the largest open set contained in S.

• A set S is open if and only if S = int(S).

• int(int(S)) = int(S) (idempotence).

• If S is a subset of T, then int(S) is a subset of int(T).

• If A is an open set, then A is a subset of S if and only if A is a subset of int(S).

Sometimes the second or third property above is taken as the definition of the topological interior.Note that these properties are also satisfied if “interior”, “subset”, “union”, “contained in”, “largest” and “open” arereplaced by “closure”, “superset”, “intersection”, “which contains”, “smallest”, and “closed”, respectively. For moreon this matter, see interior operator below.

17.2 Examples

M

( )a-ε a+εa

( [ )) [ ]

a is an interior point of M, because there is an ε-neighbourhood of a which is a subset of M.

• In any space, the interior of the empty set is the empty set.

17.3. INTERIOR OPERATOR 67

• In any space X, if A ⊂ X , int(A) is contained in A.• If X is the Euclidean space R of real numbers, then int([0, 1]) = (0, 1).• If X is the Euclidean space R , then the interior of the set Q of rational numbers is empty.• If X is the complex plane C = R2 , then int (z ∈ C : |z| ≤ 1) = z ∈ C : |z| < 1.

• In any Euclidean space, the interior of any finite set is the empty set.

On the set of real numbers one can put other topologies rather than the standard one.

• If X = R , where R has the lower limit topology, then int([0, 1]) = [0, 1).• If one considers on R the topology in which every set is open, then int([0, 1]) = [0, 1].• If one considers on R the topology in which the only open sets are the empty set and R itself, then int([0, 1])is the empty set.

These examples show that the interior of a set depends upon the topology of the underlying space. The last twoexamples are special cases of the following.

• In any discrete space, since every set is open, every set is equal to its interior.• In any indiscrete space X, since the only open sets are the empty set and X itself, we have int(X) = X and forevery proper subset A of X, int(A) is the empty set.

17.3 Interior operator

The interior operator o is dual to the closure operator —, in the sense that

So = X \ (X \ S)—,

and also

S— = X \ (X \ S)o

where X is the topological space containing S, and the backslash refers to the set-theoretic difference.Therefore, the abstract theory of closure operators and the Kuratowski closure axioms can be easily translated intothe language of interior operators, by replacing sets with their complements.

17.4 Exterior of a set

Main article: Exterior (topology)

The exterior of a subset S of a topological space X, denoted ext(S) or Ext(S), is the interior int(X \ S) of its relativecomplement. Alternatively, it can be defined as X \ S—, the complement of the closure of S. Many properties followin a straightforward way from those of the interior operator, such as the following.

• ext(S) is an open set that is disjoint with S.• ext(S) is the union of all open sets that are disjoint with S.• ext(S) is the largest open set that is disjoint with S.• If S is a subset of T, then ext(S) is a superset of ext(T).

Unlike the interior operator, ext is not idempotent, but the following holds:

• ext(ext(S)) is a superset of int(S).

68 CHAPTER 17. INTERIOR (TOPOLOGY)

17.5 Interior-disjoint shapes

The red shapes are not interior-disjoint the blue Triangle. The green and the yellow shapes are interior-disjoint with the blue Triangle,but only the yellow shape is entirely disjoint from the blue Triangle.

Two shapes a and b are called interior-disjoint if the intersection of their interiors is empty. Interior-disjoint shapesmay or may not intersect in their boundary.

17.6 See also

• Algebraic interior

• Interior algebra

• Jordan curve theorem

• Quasi-relative interior

• Relative interior

17.7 References[1] Kuratowski, Kazimierz (1922). “Sur l'Operation Ā de l'Analysis Situs” (PDF). FundamentaMathematicae (Warsaw: Polish

Academy of Sciences) 3: 182–199. ISSN 0016-2736.

17.8. EXTERNAL LINKS 69

17.8 External links• Interior at PlanetMath.org.

Chapter 18

k-cell (mathematics)

A k-cell is a higher-dimensional version of a rectangle or rectangular solid. It is the Cartesian product of k closedintervals on the real line.[1] This essentially means that is a k-dimensional rectangular solid, with each of its edgesbeing equal to one of the closed intervals used in the definition. The k intervals need not be identical. For example,a 2-cell is a rectangle in R2 such that the sides of the rectangles are parallel to the coordinate axes.

18.1 Formal definition

Let a ∈ R and b ∈ R. If ai < bi for all i = 1,...,k, the set of all points x = (x1,...,xk) in Rk whose coordinates satisfy theinequalities aᵢ ≤ x ≤ bi is a k-cell.[2] Every k-cell is compact.[3]

18.2 Intuition

A k-cell of dimension k ≤ 3 is especially simple. For example, a 1-cell is simply the interval [a,b] with a < b. A 2-cellis the rectangle formed by the Cartesian product of two closed intervals, and a 3-cell is a rectangular solid.Note that the sides and edges of a k-cell need not be equal in (Euclidean) length; although the unit cube (which hasboundaries of equal Euclidean length) is a 3-cell, the set of all 3-cells with equal-length edges is a strict subset of theset of all 3-cells.

18.3 References[1] Foran, James (1991-01-07). Fundamentals of Real Analysis. CRC Press. pp. 24–. ISBN 9780824784539. Retrieved 23

May 2014.

[2] Rudin, W: Principles of Mathematical Analysis, page 31. McGraw-Hill, 1976.

[3] Rudin, W: Principles of Mathematical Analysis, page 39. McGraw-Hill, 1976.

70

18.3. REFERENCES 71

Projections of K-cells onto the plane (from k=1 to 6. Only the edges of the higher-dimensional cells are shown.

Chapter 19

Lebesgue covering dimension

In mathematics, theLebesgue covering dimension or topological dimension of a topological space is one of severaldifferent ways of defining the dimension of the space in a topologically invariant way.

19.1 Definition

The first formal definition of covering dimension was given by Eduard Čech, based on an earlier result of HenriLebesgue.[1]

A modern definition is as follows. An open cover of a topological space X is a family of open sets whose union is X.The ply of a cover is the smallest number n (if it exists) such that each point of the space belongs to at most n setsin the cover. A refinement of a cover C is another cover, each of whose sets is a subset of a set in C; its ply may besmaller than, or possibly larger than, the ply of C. The covering dimension of a topological space X is defined to bethe minimum value of n, such that every finite open cover C of X has a refinement with ply at most n + 1. If no suchminimal n exists, the space is said to be of infinite covering dimension.As a special case, a topological space is zero-dimensional with respect to the covering dimension if every open coverof the space has a refinement consisting of disjoint open sets so that any point in the space is contained in exactly oneopen set of this refinement.

19.2 Examples

Any given open cover of the unit circle will have a refinement consisting of a collection of open arcs. The circle hasdimension one, by this definition, because any such cover can be further refined to the stage where a given point xof the circle is contained in at most two open arcs. That is, whatever collection of arcs we begin with, some can bediscarded or shrunk, such that the remainder still covers the circle but with simple overlaps.Similarly, any open cover of the unit disk in the two-dimensional plane can be refined so that any point of the diskis contained in no more than three open sets, while two are in general not sufficient. The covering dimension of thedisk is thus two.More generally, the n-dimensional Euclidean space En has covering dimension n.A non-technical illustration of these examples below.

19.3 Properties

• Homeomorphic spaces have the same covering dimension. That is, the covering dimension is a topologicalinvariant.

• The Lebesgue covering dimension coincides with the affine dimension of a finite simplicial complex; this is theLebesgue covering theorem.

72

19.4. SEE ALSO 73

• The covering dimension of a normal space is less than or equal to the large inductive dimension.

• Covering dimension of a normal space X is ≤ n if and only if for any closed subset A of X, if f : A→ Sn iscontinuous, then there is an extension of f to g : X → Sn . Here, Sn is the n dimensional sphere.

• (Ostrand’s theorem on colored dimension.) A normal space X satisfies the inequality 0 ≤ dimX ≤ n if andonly if for every locally finite open cover U = Uαα∈A of the space X there exists an open cover V of thespace X which can be represented as the union of n+ 1 families V1,V2, . . . ,Vn+1 , where Vi = Vi,αα∈A, such that each Vi contains disjoint sets and Vi,α ⊂ Uα for each i and α .

• The covering dimension of a paracompact Hausdorff spaceX is greater or equal to its cohomological dimension(in the sense of sheaves),[2] that is one hasHi(X,A) = 0 for every sheaf A of abelian groups onX and everyi larger than the covering dimension of X .

19.4 See also• Dimension theory

• Metacompact space

• Point-finite collection

19.5 Further reading

19.5.1 Historical

• Karl Menger, General Spaces and Cartesian Spaces, (1926) Communications to the Amsterdam Academy ofSciences. English translation reprinted in Classics on Fractals, Gerald A.Edgar, editor, Addison-Wesley (1993)ISBN 0-201-58701-7

• Karl Menger, Dimensionstheorie, (1928) B.G Teubner Publishers, Leipzig.

• A. R. Pears, Dimension Theory of General Spaces, (1975) Cambridge University Press. ISBN 0-521-20515-8

19.5.2 Modern

• V.V. Fedorchuk, The Fundamentals of Dimension Theory, appearing in Encyclopaedia of Mathematical Sci-ences, Volume 17, General Topology I, (1993) A. V. Arkhangel’skii and L. S. Pontryagin (Eds.), Springer-Verlag, Berlin ISBN 3-540-18178-4.

19.6 References[1] Kuperberg, Krystyna, ed. (1995), Collected Works of Witold Hurewicz, American Mathematical Society, Collected works

series 4, American Mathematical Society, p. xxiii, footnote 3, ISBN 9780821800119, Lebesgue’s discovery led later tothe introduction by E. Čech of the covering dimension.

[2] Godement 1973, II.5.12, p. 236

• Godement, Roger (1973), Topologie algébrique et théorie des faisceaux, Paris: Hermann, MR 0345092

19.7 External links• Hazewinkel, Michiel, ed. (2001), “Lebesgue dimension”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Chapter 20

Limit point compact

In mathematics, a topological space X is said to be limit point compact[1] or weakly countably compact if everyinfinite subset of X has a limit point in X. This property generalizes a property of compact spaces. In a metric space,limit point compactness, compactness, and sequential compactness are all equivalent. For general topological spaces,however, these three notions of compactness are not equivalent.

20.1 Properties and Examples• Limit point compactness is equivalent to countable compactness ifX is a T1-space and is equivalent to compactnessif X is a metric space.

• An example of a space X that is not weakly countably compact is any countable (or larger) set with the discretetopology. A more interesting example is the countable complement topology.

• Even though a continuous function from a compact space X, to an ordered set Y in the order topology, mustbe bounded, the same thing does not hold if X is limit point compact. An example is given by the spaceX ×Z(where X = 1, 2 carries the indiscrete topology and Z is the set of all integers carrying the discrete topology)and the function f = πZ given by projection onto the second coordinate. Clearly, ƒ is continuous and X × Zis limit point compact (in fact, every nonempty subset ofX ×Z has a limit point) but ƒ is not bounded, and infact f(X × Z) = Z is not even limit point compact.

• Every countably compact space (and hence every compact space) is weakly countably compact, but the converseis not true.

• For metrizable spaces, compactness, limit point compactness, and sequential compactness are all equivalent.

• The set of all real numbers, R, is not limit point compact; the integers are an infinite set but do not have a limitpoint in R.

• If (X, T) and (X, T*) are topological spaces with T* finer than T and (X, T*) is limit point compact, then so is(X, T).

• A finite space is vacuously limit point compact.

20.2 See also• Compact space

• Sequential compactness

74

20.3. NOTES 75

• Metric space

• Bolzano-Weierstrass theorem

• Countably compact space

20.3 Notes[1] The terminology “limit point compact” appears in a topology textbook by James Munkres, and is apparently due to him.

According to him, some call the property "Fréchet compactness”, while others call it the "Bolzano-Weierstrass property".Munkres, p. 178–179.

20.4 References• James Munkres (1999). Topology (2nd edition ed.). Prentice Hall. ISBN 0-13-181629-2.

• This article incorporates material from Weakly countably compact on PlanetMath, which is licensed under theCreative Commons Attribution/Share-Alike License.

Chapter 21

Lindelöf space

In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. TheLindelöf property is a weakening of the more commonly used notion of compactness, which requires the existenceof a finite subcover.A strongly Lindelöf space is a topological space such that every open subspace is Lindelöf. Such spaces are alsoknown as hereditarily Lindelöf spaces, because all subspaces of such a space are Lindelöf.Lindelöf spaces are named for the Finnish mathematician Ernst Leonard Lindelöf.

21.1 Properties of Lindelöf spaces

In general, no implications hold (in either direction) between the Lindelöf property and other compactness properties,such as paracompactness. But by the Morita theorem, every regular Lindelöf space is paracompact.Any second-countable space is a Lindelöf space, but not conversely. However, the matter is simpler for metric spaces.A metric space is Lindelöf if and only if it is separable, and if and only if it is second-countable.An open subspace of a Lindelöf space is not necessarily Lindelöf. However, a closed subspace must be Lindelöf.Lindelöf is preserved by continuous maps. However, it is not necessarily preserved by products, not even by finiteproducts.A Lindelöf space is compact if and only if it is countably compact.Any σ-compact space is Lindelöf.

21.2 Properties of strongly Lindelöf spaces• Any second-countable space is a strongly Lindelöf space

• Any Suslin space is strongly Lindelöf.

• Strongly Lindelöf spaces are closed under taking countable unions, subspaces, and continuous images.

• Every Radon measure on a strongly Lindelöf space is moderated.

21.3 Product of Lindelöf spaces

The product of Lindelöf spaces is not necessarily Lindelöf. The usual example of this is the Sorgenfrey plane S ,which is the product of the real line R under the half-open interval topology with itself. Open sets in the Sorgenfreyplane are unions of half-open rectangles that include the south and west edges and omit the north and east edges,including the northwest, northeast, and southeast corners. The antidiagonal of S is the set of points (x, y) such thatx+ y = 0 .

76

21.4. GENERALISATION 77

Consider the open covering of S which consists of:

1. The set of all rectangles (−∞, x)× (−∞, y) , where (x, y) is on the antidiagonal.

2. The set of all rectangles [x,+∞)× [y,+∞) , where (x, y) is on the antidiagonal.

The thing to notice here is that each point on the antidiagonal is contained in exactly one set of the covering, so allthese sets are needed.Another way to see that S is not Lindelöf is to note that the antidiagonal defines a closed and uncountable discretesubspace of S . This subspace is not Lindelöf, and so the whole space cannot be Lindelöf as well (as closed subspacesof Lindelöf spaces are also Lindelöf).The product of a Lindelöf space and a compact space is Lindelöf.

21.4 Generalisation

The following definition generalises the definitions of compact and Lindelöf: a topological space is κ -compact (or κ-Lindelöf), where κ is any cardinal, if every open cover has a subcover of cardinality strictly less than κ . Compact isthen ℵ0 -compact and Lindelöf is then ℵ1 -compact.The Lindelöf degree, or Lindelöf number l(X) , is the smallest cardinal κ such that every open cover of the spaceXhas a subcover of size at most κ . In this notation, X is Lindelöf if l(X) = ℵ0 . The Lindelöf number as definedabove does not distinguish between compact spaces and Lindelöf non compact spaces. Some authors gave the nameLindelöf number to a different notion: the smallest cardinal κ such that every open cover of the spaceX has a subcoverof size strictly less than κ .[1] In this latter (and less used) sense the Lindelöf number is the smallest cardinal κ suchthat a topological space X is κ -compact. This notion is sometimes also called the compactness degree of the spaceX .[2]

21.5 See also• Axioms of countability

• Lindelöf’s lemma

21.6 Notes[1] Mary Ellen Rudin, Lectures on set theoretic topology, Conference Board of the Mathematical Sciences, American Math-

ematical Society, 1975, p. 4, retrievable on Google Books

[2] Hušek,Miroslav (1969), “The class of k-compact spaces is simple”,Mathematische Zeitschrift 110: 123–126, doi:10.1007/BF01124977,MR 0244947.

21.7 References• Michael Gemignani, Elementary Topology (ISBN 0-486-66522-4) (see especially section 7.2)

• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978]. Counterexamples in Topology (Dover reprint of1978 ed.). Berlin, New York: Springer-Verlag. ISBN 978-0-486-68735-3. MR 507446.

• I. Juhász (1980). Cardinal functions in topology - ten years later. Math. Centre Tracts, Amsterdam. ISBN90-6196-196-3.

• Munkres, James. Topology, 2nd ed.

• http://arxiv.org/abs/1301.5340 Generalized Lob’s Theorem.Strong Reflection Principles and Large CardinalAxioms.Consistency Results in Topology

Chapter 22

Locally compact space

In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking,each small portion of the space looks like a small portion of a compact space.

22.1 Formal definition

Let X be a topological space. Most commonly X is called locally compact, if every point of X has a compactneighbourhood.There are other common definitions: They are all equivalent if X is a Hausdorff space (or preregular). But theyare not equivalent in general:

1. every point of X has a compact neighbourhood.2. every point of X has a closed compact neighbourhood.2′. every point of X has a relatively compact neighbourhood.2″. every point of X has a local base of relatively compact neighbourhoods.3. every point of X has a local base of compact neighbourhoods.3′. for every point x of X, every neighbourhood of x contains a compact neighbourhood of x.4. X is Hausdorff and satisfies any (all) of the previous conditions.

Logical relations among the conditions:

• Conditions (2), (2′), (2″) are equivalent.

• Conditions (3), (3′) are equivalent.

• Neither of conditions (2), (3) implies the other.

• Each condition implies (1).

• Compactness implies conditions (1) and (2), but not (3).

Condition (1) is probably the most commonly used definition, since it is the least restrictive and the others are equiv-alent to it when X is Hausdorff. This equivalence is a consequence of the facts that compact subsets of Hausdorffspaces are closed, and closed subsets of compact spaces are compact.Condition (4) is used, for example, in Bourbaki.[1] In almost all applications, locally compact spaces are indeed alsoHausdorff. These locally compact Hausdorff (LCH) spaces are thus the spaces that this article is primarily concernedwith.

78

22.2. EXAMPLES AND COUNTEREXAMPLES 79

22.2 Examples and counterexamples

22.2.1 Compact Hausdorff spaces

Every compact Hausdorff space is also locally compact, and many examples of compact spaces may be found in thearticle compact space. Here we mention only:

• the unit interval [0,1];

• the Cantor set;

• the Hilbert cube.

22.2.2 Locally compact Hausdorff spaces that are not compact

• The Euclidean spacesRn (and in particular the real lineR) are locally compact as a consequence of the Heine–Borel theorem.

• Topological manifolds share the local properties of Euclidean spaces and are therefore also all locally compact.This even includes nonparacompact manifolds such as the long line.

• All discrete spaces are locally compact and Hausdorff (they are just the zero-dimensional manifolds). Theseare compact only if they are finite.

• All open or closed subsets of a locally compact Hausdorff space are locally compact in the subspace topology.This provides several examples of locally compact subsets of Euclidean spaces, such as the unit disc (either theopen or closed version).

• The space Qp of p-adic numbers is locally compact, because it is homeomorphic to the Cantor set minus onepoint. Thus locally compact spaces are as useful in p-adic analysis as in classical analysis.

22.2.3 Hausdorff spaces that are not locally compact

Asmentioned in the following section, no Hausdorff space can possibly be locally compact if it is not also a Tychonoffspace; there are some examples of Hausdorff spaces that are not Tychonoff spaces in that article. But there are alsoexamples of Tychonoff spaces that fail to be locally compact, such as:

• the spaceQ of rational numbers (endowed with the topology fromR), since its compact subsets all have emptyinterior and therefore are not neighborhoods;

• the subspace (0,0) union (x,y) : x > 0 of R2, since the origin does not have a compact neighborhood;

• the lower limit topology or upper limit topology on the set R of real numbers (useful in the study of one-sidedlimits);

• any T0, hence Hausdorff, topological vector space that is infinite-dimensional, such as an infinite-dimensionalHilbert space.

The first two examples show that a subset of a locally compact space need not be locally compact, which contrastswith the open and closed subsets in the previous section. The last example contrasts with the Euclidean spaces inthe previous section; to be more specific, a Hausdorff topological vector space is locally compact if and only if it isfinite-dimensional (in which case it is a Euclidean space). This example also contrasts with the Hilbert cube as anexample of a compact space; there is no contradiction because the cube cannot be a neighbourhood of any point inHilbert space.

80 CHAPTER 22. LOCALLY COMPACT SPACE

22.2.4 Non-Hausdorff examples

• The one-point compactification of the rational numbers Q is compact and therefore locally compact in senses(1) and (2) but it is not locally compact in sense (3).

• The particular point topology on any infinite set is locally compact in sense (3) but not in sense (2), because ithas no nonempty closed compact subspaces containing the particular point. The same holds for the real linewith the upper topology.

22.3 Properties

Every locally compact preregular space is, in fact, completely regular. It follows that every locally compact Hausdorffspace is a Tychonoff space. Since straight regularity is a more familiar condition than either preregularity (which isusually weaker) or complete regularity (which is usually stronger), locally compact preregular spaces are normallyreferred to in the mathematical literature as locally compact regular spaces. Similarly locally compact Tychonoffspaces are usually just referred to as locally compact Hausdorff spaces.Every locally compact Hausdorff space is a Baire space. That is, the conclusion of the Baire category theorem holds:the interior of every union of countably many nowhere dense subsets is empty.A subspace X of a locally compact Hausdorff space Y is locally compact if and only if X can be written as the set-theoretic difference of two closed subsets of Y. As a corollary, a dense subspace X of a locally compact Hausdorffspace Y is locally compact if and only if X is an open subset of Y. Furthermore, if a subspace X of any Hausdorffspace Y is locally compact, then X still must be the difference of two closed subsets of Y, although the converseneedn't hold in this case.Quotient spaces of locally compact Hausdorff spaces are compactly generated. Conversely, every compactly gener-ated Hausdorff space is a quotient of some locally compact Hausdorff space.For locally compact spaces local uniform convergence is the same as compact convergence.

22.3.1 The point at infinity

Since every locally compact Hausdorff space X is Tychonoff, it can be embedded in a compact Hausdorff space b(X)using the Stone–Čech compactification. But in fact, there is a simpler method available in the locally compact case;the one-point compactification will embed X in a compact Hausdorff space a(X) with just one extra point. (The one-point compactification can be applied to other spaces, but a(X) will be Hausdorff if and only if X is locally compactand Hausdorff.) The locally compact Hausdorff spaces can thus be characterised as the open subsets of compactHausdorff spaces.Intuitively, the extra point in a(X) can be thought of as a point at infinity. The point at infinity should be thought ofas lying outside every compact subset of X. Many intuitive notions about tendency towards infinity can be formulatedin locally compact Hausdorff spaces using this idea. For example, a continuous real or complex valued function fwith domain X is said to vanish at infinity if, given any positive number e, there is a compact subset K of X suchthat |f(x)| < e whenever the point x lies outside of K. This definition makes sense for any topological space X. IfX is locally compact and Hausdorff, such functions are precisely those extendable to a continuous function g on itsone-point compactification a(X) = X ∪ ∞ where g(∞) = 0.The set C0(X) of all continuous complex-valued functions that vanish at infinity is a C* algebra. In fact, everycommutative C* algebra is isomorphic to C0(X) for some unique (up to homeomorphism) locally compact Hausdorffspace X. More precisely, the categories of locally compact Hausdorff spaces and of commutative C* algebras aredual; this is shown using the Gelfand representation. Forming the one-point compactification a(X) of X correspondsunder this duality to adjoining an identity element to C0(X).

22.3.2 Locally compact groups

The notion of local compactness is important in the study of topological groups mainly because every Hausdorfflocally compact group G carries natural measures called the Haar measures which allow one to integrate measurablefunctions defined on G. The Lebesgue measure on the real line R is a special case of this.

22.4. NOTES 81

The Pontryagin dual of a topological abelian group A is locally compact if and only if A is locally compact. Moreprecisely, Pontryagin duality defines a self-duality of the category of locally compact abelian groups. The study oflocally compact abelian groups is the foundation of harmonic analysis, a field that has since spread to non-abelianlocally compact groups.

22.4 Notes[1] Bourbaki, Nicolas (1989). General Topology, Part I (reprint of the 1966 ed.). Berlin: Springer-Verlag. ISBN 3-540-

19374-X.

22.5 References• Kelley, John (1975). General Topology. Springer. ISBN 978-0387901251.

• Munkres, James (1999). Topology (2nd ed.). Prentice Hall. ISBN 978-0131816299.

• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978]. Counterexamples in Topology (Dover reprint of1978 ed.). Berlin, New York: Springer-Verlag. ISBN 978-0-486-68735-3. MR 507446.

• Willard, Stephen (1970). General Topology. Addison-Wesley. ISBN 978-0486434797.

Chapter 23

Locally finite

The term locally finite has a number of different meanings in mathematics:

• Locally finite collection of sets in a topological space

• Locally finite group

• Locally finite measure

• Locally finite operator in linear algebra

• Locally finite poset

• Locally finite space, a topological space in which each point has a finite neighborhood

• Locally finite variety in the sense of universal algebra

82

Chapter 24

Locally finite collection

In the mathematical field of topology, local finiteness is a property of collections of subsets of a topological space.It is fundamental in the study of paracompactness and topological dimension.A collection of subsets of a topological spaceX is said to be locally finite, if each point in the space has a neighbourhoodthat intersects only finitely many of the sets in the collection.Note that the term locally finite has different meanings in other mathematical fields.

24.1 Examples and properties

A finite collection of subsets of a topological space is locally finite. Infinite collections can also be locally finite: forexample, the collection of all subsets of R of the form (n, n + 2) with integer n. A countable collection of subsetsneed not be locally finite, as shown by the collection of all subsets of R of the form (−n, n) with integer n.If a collection of sets is locally finite, the collection of all closures of these sets is also locally finite. The reasonfor this is that if an open set containing a point intersects the closure of a set, it necessarily intersects the set itself,hence a neighborhood can intersect at most the same number of closures (it may intersect fewer, since two distinct,indeed disjoint, sets can have the same closure). The converse, however, can fail if the closures of the sets are notdistinct. For example, in the finite complement topology on R the collection of all open sets is not locally finite, butthe collection of all closures of these sets is locally finite (since the only closures are R and the empty set).

24.1.1 Compact spaces

No infinite collection of a compact space can be locally finite. Indeed, let Ga be an infinite family of subsets ofa space and suppose this collection is locally finite. For each point x of this space, choose a neighbourhood Ux thatintersects the collection Ga at only finitely many values of a. Clearly:

Ux for each x in X (the union over all x) is an open covering in X

and hence has a finite subcover, Ua1 ∪ ...... ∪ Uan. Since each Uai intersects Ga for only finitely many values ofa, the union of all such Uai intersects the collection Ga for only finitely many values of a. It follows that X (thewhole space!) intersects the collection Ga at only finitely many values of a, contradicting the infinite cardinality ofthe collection Ga.A topological space in which every open cover admits a locally finite open refinement is called paracompact. Everylocally finite collection of subsets of a topological space X is also point-finite. A topological space in which everyopen cover admits a point-finite open refinement is called metacompact.

24.1.2 Second countable spaces

No uncountable cover of a Lindelöf space can be locally finite, by essentially the same argument as in the case ofcompact spaces. In particular, no uncountable cover of a second-countable space is locally finite.

83

84 CHAPTER 24. LOCALLY FINITE COLLECTION

24.2 Closed sets

It is clear from the definition of a topology that a finite union of closed sets is closed. One can readily give an exampleof an infinite union of closed sets that is not closed. However, if we consider a locally finite collection of closed sets,the union is closed. To see this we note that if x is a point outside the union of this locally finite collection of closedsets, we merely choose a neighbourhoodV of x that intersects this collection at only finitely many of these sets. Definea bijective map from the collection of sets that V intersects to 1, ..., k thus giving an index to each of these sets.Then for each set, choose an open set Ui containing x that doesn't intersect it. The intersection of all such Ui for 1 ≤i ≤ k intersected with V, is a neighbourhood of x that does not intersect the union of this collection of closed sets.

24.3 Countably locally finite collections

A collection in a space is countably locally finite (or σ-locally finite) if it is the union of a countable family of locallyfinite collections of subsets of X. Countable local finiteness is a key hypothesis in the Nagata–Smirnov metrizationtheorem, which states that a topological space is metrizable if and only if it is regular and has a countably locally finitebasis.

24.4 References• James R. Munkres (2000), Topology (2nd ed.), Prentice Hall, ISBN 0-13-181629-2

Chapter 25

Locally finite space

In the mathematical field of topology, a locally finite space is a topological space in which every point has a finiteneighborhood.A locally finite space is Alexandrov.A T1 space is locally finite if and only if it is discrete.

25.1 References• Nakaoka, Fumie; Oda, Nobuyuki (2001), “Some applications of minimal open sets”, International Journal ofMathematics and Mathematical Sciences 29 (8): 471–476, doi:10.1155/S0161171201006482

85

Chapter 26

Manifold

For other uses, see Manifold (disambiguation).In mathematics, a manifold is a topological space that resembles Euclidean space near each point. More precisely,

The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shownhere as Boy’s surface.

each point of an n-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of di-

86

26.1. MOTIVATIONAL EXAMPLES 87

The surface of the Earth requires (at least) two charts to include every point. Here the globe is decomposed into charts around theNorth and South Poles.

mension n. Lines and circles, but not figure eights, are one-dimensional manifolds. Two-dimensional manifolds arealso called surfaces. Examples include the plane, the sphere, and the torus, which can all be embedded in threedimensional real space, but also the Klein bottle and real projective plane which cannot.Although a manifold resembles Euclidean space near each point, globally it may not. For example, the surface ofthe sphere is not a Euclidean space, but in a region it can be charted by means of map projections of the region intothe Euclidean plane (in the context of manifolds they are called charts). When a region appears in two neighbouringcharts, the two representations do not coincide exactly and a transformation is needed to pass from one to the other,called a transition map.The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allowsmore complicated structures to be described and understood in terms of the relatively well-understood propertiesof Euclidean space. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions.Manifolds may have additional features. One important class of manifolds is the class of differentiable manifolds.This differentiable structure allows calculus to be done on manifolds. A Riemannian metric on a manifold allowsdistances and angles to be measured. Symplectic manifolds serve as the phase spaces in the Hamiltonian formalismof classical mechanics, while four-dimensional Lorentzian manifolds model spacetime in general relativity.

26.1 Motivational examples

26.1.1 Circle

Main article: CircleAfter a line, the circle is the simplest example of a topological manifold. Topology ignores bending, so a small pieceof a circle is treated exactly the same as a small piece of a line. Consider, for instance, the top part of the unit circle,x2 + y2 = 1, where the y-coordinate is positive (indicated by the yellow circular arc in Figure 1). Any point of this arccan be uniquely described by its x-coordinate. So, projection onto the first coordinate is a continuous, and invertible,mapping from the upper arc to the open interval (−1,1):

χtop(x, y) = x.

Such functions along with the open regions they map are called charts. Similarly, there are charts for the bottom(red), left (blue), and right (green) parts of the circle:

88 CHAPTER 26. MANIFOLD

Figure 1: The four charts each map part of the circle to an open interval, and together cover the whole circle.

χbottom(x, y) = x

χleft(x, y) = y

χright(x, y) = y.

Together, these parts cover the whole circle and the four charts form an atlas for the circle.The top and right charts overlap: their intersection lies in the quarter of the circle where both the x- and the y-coordinates are positive. The two charts χ ₒ and χᵣᵢ each map this part into the interval (0, 1). Thus a functionT from (0, 1) to itself can be constructed, which first uses the inverse of the top chart to reach the circle and thenfollows the right chart back to the interval. Let a be any number in (0, 1), then:

T (a) = χright(χ−1top [a]

)= χright

(a,√1− a2

)=√

1− a2

26.1. MOTIVATIONAL EXAMPLES 89

1

1

3

0.960.72

1.60.8

0

34

13

13

12

Figure 2: A circle manifold chart based on slope, covering all but one point of the circle.

Such a function is called a transition map.The top, bottom, left, and right charts show that the circle is a manifold, but they do not form the only possible atlas.Charts need not be geometric projections, and the number of charts is a matter of some choice. Consider the charts

χminus(x, y) = s =y

1 + x

and

χplus(x, y) = t =y

1− x

Here s is the slope of the line through the point at coordinates (x,y) and the fixed pivot point (−1, 0); t follows similarly,but with pivot point (+1, 0). The inverse mapping from s to (x, y) is given by

x =1− s2

1 + s2

y =2s

1 + s2

90 CHAPTER 26. MANIFOLD

It can easily be confirmed that x2 + y2 = 1 for all values of the slope s. These two charts provide a second atlas forthe circle, with

t =1

s

Each chart omits a single point, either (−1, 0) for s or (+1, 0) for t, so neither chart alone is sufficient to cover thewhole circle. It can be proved that it is not possible to cover the full circle with a single chart. For example, although itis possible to construct a circle from a single line interval by overlapping and “gluing” the ends, this does not producea chart; a portion of the circle will be mapped to both ends at once, losing invertibility.

26.1.2 Other curves

Four manifolds from algebraic curves:circles, parabola, hyperbola, cubic.

Manifolds need not be connected (all in “one piece”); an example is a pair of separate circles.Manifolds need not be closed; thus a line segment without its end points is a manifold. And they are never countable,unless the dimension of the manifold is 0. Putting these freedoms together, other examples of manifolds are a

26.2. HISTORY 91

parabola, a hyperbola (two open, infinite pieces), and the locus of points on a cubic curve y2 = x3−x (a closed looppiece and an open, infinite piece).However, excluded are examples like two touching circles that share a point to form a figure-8; at the shared pointa satisfactory chart cannot be created. Even with the bending allowed by topology, the vicinity of the shared pointlooks like a "+", not a line. A "+" is not homeomorphic to a closed interval (line segment), since deleting the centerpoint from the "+" gives a space with four components (i.e. pieces), whereas deleting a point from a closed intervalgives a space with at most two pieces; topological operations always preserve the number of pieces.

26.1.3 Enriched circle

Viewed using calculus, the circle transition function T is simply a function between open intervals, which gives ameaning to the statement that T is differentiable. The transition map T, and all the others, are differentiable on (0,1); therefore, with this atlas the circle is a differentiable manifold. It is also smooth and analytic because the transitionfunctions have these properties as well.Other circle properties allow it to meet the requirements of more specialized types of manifold. For example, thecircle has a notion of distance between two points, the arc-length between the points; hence it is a Riemannianmanifold.

26.2 History

For more details on this topic, see History of manifolds and varieties.

The study of manifolds combines many important areas of mathematics: it generalizes concepts such as curves andsurfaces as well as ideas from linear algebra and topology.

26.2.1 Early development

Before the modern concept of a manifold there were several important results.Non-Euclidean geometry considers spaces where Euclid's parallel postulate fails. Saccheri first studied them in 1733.Lobachevsky, Bolyai, and Riemann developed them 100 years later. Their research uncovered two types of spaceswhose geometric structures differ from that of classical Euclidean space; these gave rise to hyperbolic geometryand elliptic geometry. In the modern theory of manifolds, these notions correspond to Riemannian manifolds withconstant negative and positive curvature, respectively.Carl Friedrich Gauss may have been the first to consider abstract spaces as mathematical objects in their own right.His theorema egregium gives a method for computing the curvature of a surface without considering the ambientspace in which the surface lies. Such a surface would, in modern terminology, be called a manifold; and in modernterms, the theorem proved that the curvature of the surface is an intrinsic property. Manifold theory has come tofocus exclusively on these intrinsic properties (or invariants), while largely ignoring the extrinsic properties of theambient space.Another, more topological example of an intrinsic property of a manifold is its Euler characteristic. Leonhard Eulershowed that for a convex polytope in the three-dimensional Euclidean space with V vertices (or corners), E edges,and F faces,

V − E + F = 2.

The same formula will hold if we project the vertices and edges of the polytope onto a sphere, creating a topologicalmap with V vertices, E edges, and F faces, and in fact, will remain true for any spherical map, even if it does notarise from any convex polytope.[1] Thus 2 is a topological invariant of the sphere, called its Euler characteristic. Onthe other hand, a torus can be sliced open by its 'parallel' and 'meridian' circles, creating a map with V = 1 vertex,E = 2 edges, and F = 1 face. Thus the Euler characteristic of the torus is 1 − 2 + 1 = 0. The Euler characteristic ofother surfaces is a useful topological invariant, which can be extended to higher dimensions using Betti numbers. Inthe mid nineteenth century, the Gauss–Bonnet theorem linked the Euler characteristic to the Gaussian curvature.

92 CHAPTER 26. MANIFOLD

26.2.2 Synthesis

Investigations of Niels Henrik Abel and Carl Gustav Jacobi on inversion of elliptic integrals in the first half of 19thcentury led them to consider special types of complex manifolds, now known as Jacobians. Bernhard Riemann furthercontributed to their theory, clarifying the geometric meaning of the process of analytic continuation of functions ofcomplex variables.Another important source of manifolds in 19th century mathematics was analytical mechanics, as developed bySiméon Poisson, Jacobi, and William Rowan Hamilton. The possible states of a mechanical system are thought tobe points of an abstract space, phase space in Lagrangian and Hamiltonian formalisms of classical mechanics. Thisspace is, in fact, a high-dimensional manifold, whose dimension corresponds to the degrees of freedom of the systemand where the points are specified by their generalized coordinates. For an unconstrained movement of free particlesthe manifold is equivalent to the Euclidean space, but various conservation laws constrain it to more complicatedformations, e.g. Liouville tori. The theory of a rotating solid body, developed in the 18th century by Leonhard Eulerand Joseph-Louis Lagrange, gives another example where the manifold is nontrivial. Geometrical and topologicalaspects of classical mechanics were emphasized by Henri Poincaré, one of the founders of topology.Riemann was the first one to do extensive work generalizing the idea of a surface to higher dimensions. The namemanifold comes from Riemann’s original German term,Mannigfaltigkeit, which William Kingdon Clifford translatedas “manifoldness”. In his Göttingen inaugural lecture, Riemann described the set of all possible values of a variablewith certain constraints as a Mannigfaltigkeit, because the variable can have many values. He distinguishes betweenstetige Mannigfaltigkeit and diskrete Mannigfaltigkeit (continuous manifoldness and discontinuous manifoldness), de-pending on whether the value changes continuously or not. As continuous examples, Riemann refers to not onlycolors and the locations of objects in space, but also the possible shapes of a spatial figure. Using induction, Riemannconstructs an n-fach ausgedehnte Mannigfaltigkeit (n times extended manifoldness or n-dimensional manifoldness) asa continuous stack of (n−1) dimensional manifoldnesses. Riemann’s intuitive notion of a Mannigfaltigkeit evolvedinto what is today formalized as a manifold. Riemannian manifolds and Riemann surfaces are named after Riemann.

26.2.3 Poincaré's definition

In his very influential paper, Analysis Situs,[2] Henri Poincaré gave a definition of a (differentiable) manifold (variété)which served as a precursor to the modern concept of a manifold.[3]

In the first section of Analysis Situs, Poincaré defines a manifold as the level set of a continuously differentiablefunction between Euclidean spaces that satisfies the nondegeneracy hypothesis of the implicit function theorem. Inthe third section, he begins by remarking that the graph of a continuously differentiable function is a manifold in thelatter sense. He then proposes a new, more general, definition of manifold based on a 'chain of manifolds’ (une chaînedes variétés).Poincaré's notion of a 'chain of manifolds’ is a precursor to the modern notion of atlas. In particular, he considerstwo manifolds defined respectively as graphs of functions θ(y) and θ′(y′) . If these manifolds overlap (a une partiecommune), then he requires that the coordinates y depend continuously differentiably on the coordinates y′ and viceversa ('...les y sont fonctions analytiques des y′ et inversement'). In this way he introduces a precursor to the notionof a chart and of a transition map. Note that it is implicit in Analysis Situs that a manifold obtained as a 'chain' is asubset of Euclidean space.For example, the unit circle in the plane can be thought of as the graph of the function y =

√1− x2 or else the

function y = −√1− x2 in a neighborhood of every point except the points (1,0) and (−1,0); and in a neighborhood

of those points, it can be thought of as the graph of, respectively, x =√

1− y2 and x = −√1− y2 . The reason

the circle can be represented by a graph in the neighborhood of every point is because the left hand side of its definingequation x2 + y2 − 1 = 0 has nonzero gradient at every point of the circle. By the implicit function theorem, everysubmanifold of Euclidean space is locally the graph of a function.Hermann Weyl gave an intrinsic definition for differentiable manifolds in his lecture course on Riemann surfaces in1911–1912, opening the road to the general concept of a topological space that followed shortly. During the 1930sHassler Whitney and others clarified the foundational aspects of the subject, and thus intuitions dating back to thelatter half of the 19th century became precise, and developed through differential geometry and Lie group theory.Notably, the Whitney embedding theorem[4] showed that the intrinsic definition in terms of charts was equivalent toPoincaré's definition in terms of subsets of Euclidean space.

26.3. MATHEMATICAL DEFINITION 93

26.2.4 Topology of manifolds: highlights

Two-dimensional manifolds, also known as a 2D surfaces embedded in our common 3D space, were consideredby Riemann under the guise of Riemann surfaces, and rigorously classified in the beginning of the 20th century byPoul Heegaard and Max Dehn. Henri Poincaré pioneered the study of three-dimensional manifolds and raised afundamental question about them, today known as the Poincaré conjecture. After nearly a century of effort by manymathematicians, starting with Poincaré himself, a consensus among experts (as of 2006) is that Grigori Perelmanhas proved the Poincaré conjecture (see the Solution of the Poincaré conjecture). William Thurston's geometrizationprogram, formulated in the 1970s, provided a far-reaching extension of the Poincaré conjecture to the general three-dimensional manifolds. Four-dimensional manifolds were brought to the forefront of mathematical research in the1980s by Michael Freedman and in a different setting, by Simon Donaldson, who was motivated by the then recentprogress in theoretical physics (Yang–Mills theory), where they serve as a substitute for ordinary 'flat' spacetime.Andrey Markov Jr. showed in 1960 that no algorithm exists for classifying four-dimensional manifolds. Importantwork on higher-dimensional manifolds, including analogues of the Poincaré conjecture, had been done earlier byRené Thom, John Milnor, Stephen Smale and Sergei Novikov. One of the most pervasive and flexible techniquesunderlying much work on the topology of manifolds is Morse theory.

26.3 Mathematical definition

For more details on this topic, see Categories of manifolds.

Informally, a manifold is a space that is “modeled on” Euclidean space.There are many different kinds of manifolds and generalizations. In geometry and topology, all manifolds aretopological manifolds, possibly with additional structure, most often a differentiable structure. In terms of construct-ing manifolds via patching, a manifold has an additional structure if the transition maps between different patchessatisfy axioms beyond just continuity. For instance, differentiable manifolds have homeomorphisms on overlappingneighborhoods diffeomorphic with each other, so that the manifold has a well-defined set of functions which aredifferentiable in each neighborhood, and so differentiable on the manifold as a whole.Formally, a topological manifold[5] is a second countable Hausdorff space that is locally homeomorphic to Euclideanspace.Second countable and Hausdorff are point-set conditions; second countable excludes spaces which are in some sense'too large' such as the long line, while Hausdorff excludes spaces such as “the line with two origins” (these general-izations of manifolds are discussed in non-Hausdorff manifolds).Locally homeomorphic to Euclidean space means[6] that every point has a neighborhood homeomorphic to an openEuclidean n-ball,

Bn = (x1, x2, . . . , xn) ∈ Rn | x21 + x22 + · · ·+ x2n < 1.

Generally manifolds are taken to have a fixed dimension (the space must be locally homeomorphic to a fixed n-ball),and such a space is called an n-manifold; however, some authors admit manifolds where different points can havedifferent dimensions.[7] If a manifold has a fixed dimension, it is called a pure manifold. For example, the spherehas a constant dimension of 2 and is therefore a pure manifold whereas the disjoint union of a sphere and a line inthree-dimensional space is not a pure manifold. Since dimension is a local invariant (i.e. the map sending each pointto the dimension of its neighbourhood over which a chart is defined, is locally constant), each connected componenthas a fixed dimension.Scheme-theoretically, a manifold is a locally ringed space, whose structure sheaf is locally isomorphic to the sheaf ofcontinuous (or differentiable, or complex-analytic, etc.) functions on Euclidean space. This definition is mostly usedwhen discussing analytic manifolds in algebraic geometry.

26.3.1 Broad definition

Main article: Banach manifold

94 CHAPTER 26. MANIFOLD

The broadest common definition of manifold is a topological space locally homeomorphic to a topological vectorspace over the reals. This omits the point-set axioms, allowing higher cardinalities and non-Hausdorff manifolds;and it omits finite dimension, allowing structures such as Hilbert manifolds to be modeled on Hilbert spaces, Banachmanifolds to be modeled on Banach spaces, and Fréchet manifolds to be modeled on Fréchet spaces. Usually onerelaxes one or the other condition: manifolds with the point-set axioms are studied in general topology, while infinite-dimensional manifolds are studied in functional analysis.

26.4 Charts, atlases, and transition maps

Main article: Atlas (topology)See also: Differentiable manifold

The spherical Earth is navigated using flat maps or charts, collected in an atlas. Similarly, a differentiable manifold canbe described using mathematical maps, called coordinate charts, collected in a mathematical atlas. It is not generallypossible to describe a manifold with just one chart, because the global structure of the manifold is different fromthe simple structure of the charts. For example, no single flat map can represent the entire Earth without separationof adjacent features across the map’s boundaries or duplication of coverage. When a manifold is constructed frommultiple overlapping charts, the regions where they overlap carry information essential to understanding the globalstructure.

26.4.1 Charts

A coordinate map, a coordinate chart, or simply a chart, of a manifold is an invertible map between a subsetof the manifold and a simple space such that both the map and its inverse preserve the desired structure.[8] For atopological manifold, the simple space is some Euclidean space Rn and interest focuses on the topological structure.This structure is preserved by homeomorphisms, invertible maps that are continuous in both directions.In the case of a differentiable manifold, a set of charts called an atlas allows us to do calculus on manifolds. Polarcoordinates, for example, form a chart for the plane R2 minus the positive x-axis and the origin. Another example ofa chart is the map χ ₒ mentioned in the section above, a chart for the circle.

26.4.2 Atlases

Main article: Atlas (topology)

The description of most manifolds requires more than one chart (a single chart is adequate for only the simplestmanifolds). A specific collection of charts which covers a manifold is called an atlas. An atlas is not unique asall manifolds can be covered multiple ways using different combinations of charts. Two atlases are said to be Ck-equivalent if their union is also a Ck atlas.The atlas containing all possible charts consistent with a given atlas is called the maximal atlas (i.e. an equivalenceclass containing that given atlas (under the already defined equivalence relation given in the previous paragraph)).Unlike an ordinary atlas, the maximal atlas of a given manifold is unique. Though it is useful for definitions, it is anabstract object and not used directly (e.g. in calculations).

26.4.3 Transition maps

Charts in an atlas may overlap and a single point of a manifold may be represented in several charts. If two chartsoverlap, parts of them represent the same region of the manifold, just as a map of Europe and a map of Asia mayboth contain Moscow. Given two overlapping charts, a transition function can be defined which goes from an openball inRn to the manifold and then back to another (or perhaps the same) open ball inRn. The resultant map, like themap T in the circle example above, is called a change of coordinates, a coordinate transformation, a transitionfunction, or a transition map.

26.5. MANIFOLD WITH BOUNDARY 95

26.4.4 Additional structure

An atlas can also be used to define additional structure on the manifold. The structure is first defined on each chartseparately. If all the transition maps are compatible with this structure, the structure transfers to the manifold.This is the standard way differentiable manifolds are defined. If the transition functions of an atlas for a topologi-cal manifold preserve the natural differential structure of Rn (that is, if they are diffeomorphisms), the differentialstructure transfers to the manifold and turns it into a differentiable manifold. Complex manifolds are introduced inan analogous way by requiring that the transition functions of an atlas are holomorphic functions. For symplecticmanifolds, the transition functions must be symplectomorphisms.The structure on the manifold depends on the atlas, but sometimes different atlases can be said to give rise to thesame structure. Such atlases are called compatible.These notions are made precise in general through the use of pseudogroups.

26.5 Manifold with boundary

See also: Topological manifold § Manifolds with boundary

A manifold with boundary is a manifold with an edge. For example a sheet of paper is a 2-manifold with a 1-dimensional boundary. The boundary of an n-manifold with boundary is an (n − 1)-manifold. A disk (circle plusinterior) is a 2-manifold with boundary. Its boundary is a circle, a 1-manifold. A square with interior is also a 2-manifold with boundary. A ball (sphere plus interior) is a 3-manifold with boundary. Its boundary is a sphere, a2-manifold. (See also Boundary (topology)).In technical language, a manifold with boundary is a space containing both interior points and boundary points.Every interior point has a neighborhood homeomorphic to the open n-ball (x1, x2, …, xn) | Σ xi2 < 1. Everyboundary point has a neighborhood homeomorphic to the “half” n-ball (x1, x2, …, xn) | Σ xi2 < 1 and x1 ≥ 0. Thehomeomorphism must send each boundary point to a point with x1 = 0.

26.5.1 Boundary and interior

Let M be a manifold with boundary. The interior of M, denoted Int M, is the set of points in M which haveneighborhoods homeomorphic to an open subset of Rn. The boundary ofM, denoted ∂M, is the complement of IntM in M. The boundary points can be characterized as those points which land on the boundary hyperplane (xn = 0)of Rn₊ under some coordinate chart.If M is a manifold with boundary of dimension n, then Int M is a manifold (without boundary) of dimension n and∂M is a manifold (without boundary) of dimension n − 1.

26.6 Construction

A single manifold can be constructed in different ways, each stressing a different aspect of the manifold, therebyleading to a slightly different viewpoint.

26.6.1 Charts

Perhaps the simplest way to construct a manifold is the one used in the example above of the circle. First, a subsetof R2 is identified, and then an atlas covering this subset is constructed. The concept of manifold grew historicallyfrom constructions like this. Here is another example, applying this method to the construction of a sphere:

Sphere with charts

A sphere can be treated in almost the same way as the circle. In mathematics a sphere is just the surface (not thesolid interior), which can be defined as a subset of R3:

96 CHAPTER 26. MANIFOLD

S = (x, y, z) ∈ R3|x2 + y2 + z2 = 1.

The sphere is two-dimensional, so each chart will map part of the sphere to an open subset of R2. Consider thenorthern hemisphere, which is the part with positive z coordinate (coloured red in the picture on the right). Thefunction χ defined by

χ(x, y, z) = (x, y),

maps the northern hemisphere to the open unit disc by projecting it on the (x, y) plane. A similar chart exists for thesouthern hemisphere. Together with two charts projecting on the (x, z) plane and two charts projecting on the (y, z)plane, an atlas of six charts is obtained which covers the entire sphere.This can be easily generalized to higher-dimensional spheres.

26.6.2 Patchwork

A manifold can be constructed by gluing together pieces in a consistent manner, making them into overlappingcharts. This construction is possible for any manifold and hence it is often used as a characterisation, especially fordifferentiable and Riemannian manifolds. It focuses on an atlas, as the patches naturally provide charts, and sincethere is no exterior space involved it leads to an intrinsic view of the manifold.Themanifold is constructed by specifying an atlas, which is itself defined by transitionmaps. A point of themanifold istherefore an equivalence class of points which are mapped to each other by transition maps. Charts map equivalenceclasses to points of a single patch. There are usually strong demands on the consistency of the transition maps.For topological manifolds they are required to be homeomorphisms; if they are also diffeomorphisms, the resultingmanifold is a differentiable manifold.This can be illustrated with the transition map t = 1⁄s from the second half of the circle example. Start with two copiesof the line. Use the coordinate s for the first copy, and t for the second copy. Now, glue both copies together byidentifying the point t on the second copy with the point s = 1⁄t on the first copy (the points t = 0 and s = 0 are notidentified with any point on the first and second copy, respectively). This gives a circle.

Intrinsic and extrinsic view

The first construction and this construction are very similar, but they represent rather different points of view. In thefirst construction, the manifold is seen as embedded in some Euclidean space. This is the extrinsic view. When amanifold is viewed in this way, it is easy to use intuition from Euclidean spaces to define additional structure. Forexample, in a Euclidean space it is always clear whether a vector at some point is tangential or normal to some surfacethrough that point.The patchwork construction does not use any embedding, but simply views the manifold as a topological space byitself. This abstract point of view is called the intrinsic view. It can make it harder to imagine what a tangent vectormight be, and there is no intrinsic notion of a normal bundle, but instead there is an intrinsic stable normal bundle.

n-Sphere as a patchwork

The n-sphere Sn is a generalisation of the idea of a circle (1-sphere) and sphere (2-sphere) to higher dimensions. Ann-sphere Sn can be constructed by gluing together two copies of Rn. The transition map between them is defined as

Rn \ 0 → Rn \ 0 : x 7→ x/∥x∥2.

This function is its own inverse and thus can be used in both directions. As the transition map is a smooth function,this atlas defines a smooth manifold. In the case n = 1, the example simplifies to the circle example given earlier.

26.7. MANIFOLDS WITH ADDITIONAL STRUCTURE 97

26.6.3 Identifying points of a manifold

Main articles: Orbifold and Group action

It is possible to define different points of a manifold to be same. This can be visualized as gluing these pointstogether in a single point, forming a quotient space. There is, however, no reason to expect such quotient spaces tobe manifolds. Among the possible quotient spaces that are not necessarily manifolds, orbifolds and CW complexesare considered to be relatively well-behaved. An example of a quotient space of a manifold that is also a manifold isthe real projective space identified as a quotient space of the corresponding sphere.One method of identifying points (gluing them together) is through a right (or left) action of a group, which acts onthe manifold. Two points are identified if one is moved onto the other by some group element. If M is the manifoldand G is the group, the resulting quotient space is denoted by M / G (or G \ M).Manifolds which can be constructed by identifying points include tori and real projective spaces (starting with a planeand a sphere, respectively).

26.6.4 Gluing along boundaries

Main article: Quotient space (topology)

Two manifolds with boundaries can be glued together along a boundary. If this is done the right way, the result isalso a manifold. Similarly, two boundaries of a single manifold can be glued together.Formally, the gluing is defined by a bijection between the two boundaries. Two points are identified when they aremapped onto each other. For a topological manifold this bijection should be a homeomorphism, otherwise the resultwill not be a topological manifold. Similarly for a differentiable manifold it has to be a diffeomorphism. For othermanifolds other structures should be preserved.A finite cylinder may be constructed as a manifold by starting with a strip [0, 1] × [0, 1] and gluing a pair of oppositeedges on the boundary by a suitable diffeomorphism. A projective plane may be obtained by gluing a sphere with ahole in it to a Möbius strip along their respective circular boundaries.

26.6.5 Cartesian products

The Cartesian product of manifolds is also a manifold.The dimension of the product manifold is the sum of the dimensions of its factors. Its topology is the product topology,and a Cartesian product of charts is a chart for the product manifold. Thus, an atlas for the product manifold can beconstructed using atlases for its factors. If these atlases define a differential structure on the factors, the correspondingatlas defines a differential structure on the product manifold. The same is true for any other structure defined on thefactors. If one of the factors has a boundary, the product manifold also has a boundary. Cartesian products may beused to construct tori and finite cylinders, for example, as S1 × S1 and S1 × [0, 1], respectively.

26.7 Manifolds with additional structure

Main article: Categories of manifolds

26.7.1 Topological manifolds

Main article: topological manifold

The simplest kind ofmanifold to define is the topologicalmanifold, which looks locally like some “ordinary” Euclideanspace Rn. Formally, a topological manifold is a topological space locally homeomorphic to a Euclidean space. Thismeans that every point has a neighbourhood for which there exists a homeomorphism (a bijective continuous function

98 CHAPTER 26. MANIFOLD

whose inverse is also continuous) mapping that neighbourhood to Rn. These homeomorphisms are the charts of themanifold.It is to be noted that a topologicalmanifold looks locally like a Euclidean space in a rather weakmanner: while for eachindividual chart it is possible to distinguish differentiable functions or measure distances and angles, merely by virtueof being a topological manifold a space does not have any particular and consistent choice of such concepts. In orderto discuss such properties for a manifold, one needs to specify further structure and consider differentiable manifoldsand Riemannian manifolds discussed below. In particular, the same underlying topological manifold can have severalmutually incompatible classes of differentiable functions and an infinite number of ways to specify distances andangles.Usually additional technical assumptions on the topological space are made to exclude pathological cases. It is cus-tomary to require that the space be Hausdorff and second countable.The dimension of the manifold at a certain point is the dimension of the Euclidean space that the charts at that pointmap to (number n in the definition). All points in a connected manifold have the same dimension. Some authorsrequire that all charts of a topological manifold map to Euclidean spaces of same dimension. In that case everytopological manifold has a topological invariant, its dimension. Other authors allow disjoint unions of topologicalmanifolds with differing dimensions to be called manifolds.

26.7.2 Differentiable manifolds

Main article: Differentiable manifold

For most applications a special kind of topological manifold, namely a differentiable manifold, is used. If the localcharts on a manifold are compatible in a certain sense, one can define directions, tangent spaces, and differentiablefunctions on that manifold. In particular it is possible to use calculus on a differentiable manifold. Each point of ann-dimensional differentiable manifold has a tangent space. This is an n-dimensional Euclidean space consisting ofthe tangent vectors of the curves through the point.Two important classes of differentiable manifolds are smooth and analytic manifolds. For smooth manifolds thetransition maps are smooth, that is infinitely differentiable. Analytic manifolds are smooth manifolds with the ad-ditional condition that the transition maps are analytic (they can be expressed as power series). The sphere can begiven analytic structure, as can most familiar curves and surfaces.There are also topological manifolds, i.e., locally Euclidean spaces, which possess no differentiable structures at all.[9]

A rectifiable set generalizes the idea of a piecewise smooth or rectifiable curve to higher dimensions; however, recti-fiable sets are not in general manifolds.

26.7.3 Riemannian manifolds

Main article: Riemannian manifold

To measure distances and angles on manifolds, the manifold must be Riemannian. A 'Riemannian manifold' is adifferentiable manifold in which each tangent space is equipped with an inner product ⟨⋅,⋅⟩ in a manner which variessmoothly from point to point. Given two tangent vectors u and v, the inner product ⟨u,v⟩ gives a real number. Thedot (or scalar) product is a typical example of an inner product. This allows one to define various notions such aslength, angles, areas (or volumes), curvature, gradients of functions and divergence of vector fields.All differentiable manifolds (of constant dimension) can be given the structure of a Riemannian manifold. TheEuclidean space itself carries a natural structure of Riemannian manifold (the tangent spaces are naturally identifiedwith the Euclidean space itself and carry the standard scalar product of the space). Many familiar curves and surfaces,including for example all n-spheres, are specified as subspaces of a Euclidean space and inherit a metric from theirembedding in it.

26.8. CLASSIFICATION AND INVARIANTS 99

26.7.4 Finsler manifolds

Main article: Finsler manifold

A Finsler manifold allows the definition of distance but does not require the concept of angle; it is an analyticmanifold in which each tangent space is equipped with a norm, ||·||, in a manner which varies smoothly from pointto point. This norm can be extended to a metric, defining the length of a curve; but it cannot in general be used todefine an inner product.Any Riemannian manifold is a Finsler manifold.

26.7.5 Lie groups

Main article: Lie group

Lie groups, named after Sophus Lie, are differentiable manifolds that carry also the structure of a group which issuch that the group operations are defined by smooth maps.A Euclidean vector space with the group operation of vector addition is an example of a non-compact Lie group.A simple example of a compact Lie group is the circle: the group operation is simply rotation. This group, knownas U(1), can be also characterised as the group of complex numbers of modulus 1 with multiplication as the groupoperation. Other examples of Lie groups include special groups of matrices, which are all subgroups of the generallinear group, the group of n by n matrices with non-zero determinant. If the matrix entries are real numbers, thiswill be an n2-dimensional disconnected manifold. The orthogonal groups, the symmetry groups of the sphere andhyperspheres, are n(n−1)/2 dimensional manifolds, where n−1 is the dimension of the sphere. Further examples canbe found in the table of Lie groups.

26.7.6 Other types of manifolds

Main articles: Complex manifold and Symplectic manifold

• A 'complex manifold' is a manifold modeled on Cn with holomorphic transition functions on chart overlaps.These manifolds are the basic objects of study in complex geometry. A one-complex-dimensional manifold iscalled a Riemann surface. Note that an n-dimensional complex manifold has dimension 2n as a real differen-tiable manifold.

• A 'CR manifold' is a manifold modeled on boundaries of domains in Cn .

• 'Infinite dimensional manifolds’: to allow for infinite dimensions, one may consider Banachmanifolds which arelocally homeomorphic to Banach spaces. Similarly, Fréchet manifolds are locally homeomorphic to Fréchetspaces.

• A 'symplectic manifold' is a kind of manifold which is used to represent the phase spaces in classical mechanics.They are endowed with a 2-form that defines the Poisson bracket. A closely related type of manifold is a contactmanifold.

• A 'combinatorial manifold' is a kind of manifold which is discretization of a manifold. It usually means apiecewise linear manifold made by simplicial complexes.

• A 'digital manifold' is a special kind of combinatorial manifold which is defined in digital space. See digitaltopology

26.8 Classification and invariants

For more details on this topic, see Classification of manifolds.

100 CHAPTER 26. MANIFOLD

Different notions of manifolds have different notions of classification and invariant; in this section we focus on smoothclosed manifolds.The classification of smooth closedmanifolds is well-understood in principle, except in dimension 4: in low dimensions(2 and 3) it is geometric, via the uniformization theorem and the solution of the Poincaré conjecture, and in highdimension (5 and above) it is algebraic, via surgery theory. This is a classification in principle: the general questionof whether two smooth manifolds are diffeomorphic is not computable in general. Further, specific computationsremain difficult, and there are many open questions.Orientable surfaces can be visualized, and their diffeomorphism classes enumerated, by genus. Given two orientablesurfaces, one can determine if they are diffeomorphic by computing their respective genera and comparing: they arediffeomorphic if and only if the genera are equal, so the genus forms a complete set of invariants.This is much harder in higher dimensions: higher-dimensional manifolds cannot be directly visualized (though visualintuition is useful in understanding them), nor can their diffeomorphism classes be enumerated, nor can one in generaldetermine if two different descriptions of a higher-dimensional manifold refer to the same object.However, one can determine if two manifolds are different if there is some intrinsic characteristic that differentiatesthem. Such criteria are commonly referred to as invariants, because, while they may be defined in terms of somepresentation (such as the genus in terms of a triangulation), they are the same relative to all possible descriptions ofa particular manifold: they are invariant under different descriptions.Naively, one could hope to develop an arsenal of invariant criteria that would definitively classify all manifolds up toisomorphism. Unfortunately, it is known that for manifolds of dimension 4 and higher, no program exists that candecide whether two manifolds are diffeomorphic.Smooth manifolds have a rich set of invariants, coming from point-set topology, classic algebraic topology, andgeometric topology. The most familiar invariants, which are visible for surfaces, are orientability (a normal invariant,also detected by homology) and genus (a homological invariant).Smooth closed manifolds have no local invariants (other than dimension), though geometric manifolds have localinvariants, notably the curvature of a Riemannian manifold and the torsion of a manifold equipped with an affineconnection. This distinction between local invariants and no local invariants is a common way to distinguish betweengeometry and topology. All invariants of a smooth closed manifold are thus global.Algebraic topology is a source of a number of important global invariant properties. Some key criteria include thesimply connected property and orientability (see below). Indeed several branches of mathematics, such as homologyand homotopy theory, and the theory of characteristic classes were founded in order to study invariant properties ofmanifolds.

26.9 Examples of surfaces

26.9.1 Orientability

Main article: Orientable manifold

In dimensions two and higher, a simple but important invariant criterion is the question of whether a manifold admitsa meaningful orientation. Consider a topological manifold with charts mapping to Rn. Given an ordered basis for Rn,a chart causes its piece of the manifold to itself acquire a sense of ordering, which in 3-dimensions can be viewed aseither right-handed or left-handed. Overlapping charts are not required to agree in their sense of ordering, which givesmanifolds an important freedom. For somemanifolds, like the sphere, charts can be chosen so that overlapping regionsagree on their “handedness"; these are orientable manifolds. For others, this is impossible. The latter possibility iseasy to overlook, because any closed surface embedded (without self-intersection) in three-dimensional space isorientable.Some illustrative examples of non-orientable manifolds include: (1) the Möbius strip, which is a manifold withboundary, (2) the Klein bottle, which must intersect itself in its 3-space representation, and (3) the real projectiveplane, which arises naturally in geometry.

26.10. MAPS OF MANIFOLDS 101

Möbius strip

Main article: Möbius strip

Begin with an infinite circular cylinder standing vertically, a manifold without boundary. Slice across it high andlow to produce two circular boundaries, and the cylindrical strip between them. This is an orientable manifold withboundary, upon which “surgery” will be performed. Slice the strip open, so that it could unroll to become a rectangle,but keep a grasp on the cut ends. Twist one end 180°, making the inner surface face out, and glue the ends backtogether seamlessly. This results in a strip with a permanent half-twist: the Möbius strip. Its boundary is no longera pair of circles, but (topologically) a single circle; and what was once its “inside” has merged with its “outside”, sothat it now has only a single side.

Klein bottle

Main article: Klein bottleTake twoMöbius strips; each has a single loop as a boundary. Straighten out those loops into circles, and let the stripsdistort into cross-caps. Gluing the circles together will produce a new, closed manifold without boundary, the Kleinbottle. Closing the surface does nothing to improve the lack of orientability, it merely removes the boundary. Thus,the Klein bottle is a closed surface with no distinction between inside and outside. Note that in three-dimensionalspace, a Klein bottle’s surface must pass through itself. Building a Klein bottle which is not self-intersecting requiresfour or more dimensions of space.

Real projective plane

Main article: Real projective space

Begin with a sphere centered on the origin. Every line through the origin pierces the sphere in two opposite pointscalled antipodes. Although there is no way to do so physically, it is possible (by considering a quotient space) tomathematically merge each antipode pair into a single point. The closed surface so produced is the real projectiveplane, yet another non-orientable surface. It has a number of equivalent descriptions and constructions, but this routeexplains its name: all the points on any given line through the origin project to the same “point” on this “plane”.

26.9.2 Genus and the Euler characteristic

For two dimensional manifolds a key invariant property is the genus, or the “number of handles” present in a surface.A torus is a sphere with one handle, a double torus is a sphere with two handles, and so on. Indeed it is possible tofully characterize compact, two-dimensional manifolds on the basis of genus and orientability. In higher-dimensionalmanifolds genus is replaced by the notion of Euler characteristic, and more generally Betti numbers and homologyand cohomology.

26.10 Maps of manifolds

Main article: Maps of manifolds

Just as there are various types of manifolds, there are various types of maps of manifolds. In addition to continuousfunctions and smooth functions generally, there are maps with special properties. In geometric topology a basic typeare embeddings, of which knot theory is a central example, and generalizations such as immersions, submersions,covering spaces, and ramified covering spaces. Basic results include the Whitney embedding theorem and Whitneyimmersion theorem.In Riemannian geometry, one may ask for maps to preserve the Riemannian metric, leading to notions of isometricembeddings, isometric immersions, and Riemannian submersions; a basic result is the Nash embedding theorem.

102 CHAPTER 26. MANIFOLD

26.10.1 Scalar-valued functions

A basic example of maps between manifolds are scalar-valued functions on a manifold,

f : M → R or f : M → C,

sometimes called regular functions or functionals, by analogy with algebraic geometry or linear algebra. These are ofinterest both in their own right, and to study the underlying manifold.In geometric topology, most commonly studied are Morse functions, which yield handlebody decompositions, whilein mathematical analysis, one often studies solution to partial differential equations, an important example of whichis harmonic analysis, where one studies harmonic functions: the kernel of the Laplace operator. This leads to suchfunctions as the spherical harmonics, and to heat kernel methods of studying manifolds, such as hearing the shape ofa drum and some proofs of the Atiyah–Singer index theorem.

26.11 Generalizations of manifolds• Orbifolds: An orbifold is a generalization of manifold allowing for certain kinds of "singularities" in thetopology. Roughly speaking, it is a space which locally looks like the quotients of some simple space (e.g.Euclidean space) by the actions of various finite groups. The singularities correspond to fixed points of thegroup actions, and the actions must be compatible in a certain sense.

• Algebraic varieties and schemes: Non-singular algebraic varieties over the real or complex numbers aremanifolds. One generalizes this first by allowing singularities, secondly by allowing different fields, and thirdlyby emulating the patching construction of manifolds: just as a manifold is glued together from open subsets ofEuclidean space, an algebraic variety is glued together from affine algebraic varieties, which are zero sets ofpolynomials over algebraically closed fields. Schemes are likewise glued together from affine schemes, whichare a generalization of algebraic varieties. Both are related to manifolds, but are constructed algebraically usingsheaves instead of atlases.

Because of singular points, a variety is in general not a manifold, though linguistically the French variété,German Mannigfaltigkeit and English manifold are largely synonymous. In French an algebraic varietyis called une variété algébrique (an algebraic variety), while a smooth manifold is called une variétédifférentielle (a differential variety).

• Stratified space: A “stratified space” is a space that can be divided into pieces (“strata”), with each stratum amanifold, with the strata fitting together in prescribed ways (formally, a filtration by closed subsets). There arevarious technical definitions, notably a Whitney stratified space (see Whitney conditions) for smooth manifoldsand a topologically stratified space for topological manifolds. Basic examples include manifold with boundary(top dimensional manifold and codimension 1 boundary) and manifold with corners (top dimensional mani-fold, codimension 1 boundary, codimension 2 corners). Whitney stratified spaces are a broad class of spaces,including algebraic varieties, analytic varieties, semialgebraic sets, and subanalytic sets.

• CW-complexes: A CW complex is a topological space formed by gluing disks of different dimensionalitytogether. In general the resulting space is singular, and hence not a manifold. However, they are of centralinterest in algebraic topology, especially in homotopy theory, as they are easy to compute with and singularitiesare not a concern.

• Homology manifolds: A homology manifold is a space that behaves like a manifold from the point of viewof homology theory. These are not all manifolds, but (in high dimension) can be analyzed by surgery theorysimilarly to manifolds, and failure to be a manifold is a local obstruction, as in surgery theory.[10]

• Differential spaces: LetM be a nonempty set. Suppose that some family of real functions onM was chosen.Denote it by C ⊆ RM . It is an algebra with respect to the pointwise addition and multiplication. LetM beequipped with the topology induced by C . Suppose also that the following conditions hold. First: for everyH ∈ C∞(Ri) , where i ∈ N , and arbitrary f1, . . . , fn ∈ C , the compositionH (f1, . . . , fn) ∈ C . Second:every function, which in every point ofM locally coincides with some function from C , also belongs to C .A pair (M,C) for which the above conditions hold, is called a Sikorski differential space.[11] [12]

26.12. SEE ALSO 103

26.12 See also

• Affine geodesic: paths on manifolds

• Directional statistics: statistics on manifolds

• List of manifolds

• Mathematics of general relativity

• Submanifold

26.12.1 By dimension

• Curve (1-manifold)

• Surface (2-manifold)

• 3-manifold

• 4-manifold

• 5-manifold

• Banach manifold

• Fréchet manifold

• Manifolds of mappings

26.13 Notes[1] The notion of a map can formalized as a cell decomposition.

[2] Poincaré, H.: Analysis Situs. (French) Journal de l'Ecole Polytechnique, Serié 11 Gauthier-Villars (1895).

[3] Arnolʹd, V. I.: On the teaching of mathematics.(Russian) Uspekhi Mat. Nauk 53 (1998), no. 1(319), 229–234; translationin Russian Math. Surveys 53 (1998), no. 1, 229–236

[4] Whitney H., Differentiable manifolds, Ann. of Math. (2), 37 (1936), 645–680.

[5] In the narrow sense of requiring point-set axioms and finite dimension.

[6] Formally, locally homeomorphic means that each point m in the manifold M has a neighborhood homeomorphic to aneighborhood in Euclidean space, not to the unit ball specifically. However, given such a homeomorphism, the pre-imageof an ϵ -ball gives a homeomorphism between the unit ball and a smaller neighborhood ofm, so this is no loss of generality.For topological or differentiable manifolds, one can also ask that every point have a neighborhood homeomorphic to all ofEuclidean space (as this is diffeomorphic to the unit ball), but this cannot be done for complex manifolds, as the complexunit ball is not holomorphic to complex space.

[7] E.g. see Riaza, Ricardo (2008), Differential-Algebraic Systems: Analytical Aspects and Circuit Applications, World Scien-tific, p. 110, ISBN 9789812791818; Gunning, R. C. (1990), Introduction to Holomorphic Functions of Several Variables,Volume 2, CRC Press, p. 73, ISBN 9780534133092.

[8] Shigeyuki Morita, Teruko Nagase, Katsumi Nomizu (2001). Geometry of Differential Forms. American MathematicalSociety Bookstore. p. 12. ISBN 0-8218-1045-6.

[9] Kervaire M., A Manifold which does not admit any differentiable structure, Comment. Math. Helv., 35 (1961), 1–14.

[10] J. Bryant, S. Ferry, W. Mio, and S. Weinberger, Topology of homology manifolds, Annals of Maths. 143, 435–467 (1996)

[11] R. Sikorski, Abstract covariant derivative, Coll. Math. 18, 251-272 (1967)

[12] K. Drachal, Introduction to d–spaces theory,Math. Aeterna 3, 753-770 (2013)

104 CHAPTER 26. MANIFOLD

26.14 References• Freedman, Michael H., and Quinn, Frank (1990) Topology of 4-Manifolds. Princeton University Press. ISBN0-691-08577-3.

• Guillemin, Victor and Pollack, Alan (1974) Differential Topology. Prentice-Hall. ISBN 0-13-212605-2. In-spired by Milnor and commonly used in undergraduate courses.

• Hempel, John (1976) 3-Manifolds. Princeton University Press. ISBN 0-8218-3695-1.

• Hirsch, Morris, (1997) Differential Topology. Springer Verlag. ISBN 0-387-90148-5. The most completeaccount, with historical insights and excellent, but difficult, problems. The standard reference for those wishingto have a deep understanding of the subject.

• Kirby, RobionC. and Siebenmann, LaurenceC. (1977)Foundational Essays on TopologicalManifolds. Smooth-ings, and Triangulations. Princeton University Press. ISBN 0-691-08190-5. A detailed study of the categoryof topological manifolds.

• Lee, John M. (2000) Introduction to Topological Manifolds. Springer-Verlag. ISBN 0-387-98759-2.

• Lee, John M. (2002), Introduction to Smooth Manifolds, Springer, ISBN 978-0-387-95448-6

• Lee, John M. (2003) Introduction to Smooth Manifolds. Springer-Verlag. ISBN 0-387-95495-3.

• Massey, William S. (1977) Algebraic Topology: An Introduction. Springer-Verlag. ISBN 0-387-90271-6.

• Milnor, John (1997) Topology from the Differentiable Viewpoint. Princeton University Press. ISBN 0-691-04833-9.

• Munkres, James R. (2000) Topology. Prentice Hall. ISBN 0-13-181629-2.

• Neuwirth, L. P., ed. (1975) Knots, Groups, and 3-Manifolds. Papers Dedicated to the Memory of R. H. Fox.Princeton University Press. ISBN 978-0-691-08170-0.

• Riemann, Bernhard,Gesammelte mathematischeWerke und wissenschaftlicher Nachlass, Sändig Reprint. ISBN3-253-03059-8.

• Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse. The 1851doctoral thesis in which “manifold” (Mannigfaltigkeit) first appears.

• Ueber die Hypothesen, welche der Geometrie zu Grunde liegen. The 1854 Göttingen inaugural lecture(Habilitationsschrift).

• Spivak, Michael (1965) Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Cal-culus. HarperCollins Publishers. ISBN 0-8053-9021-9. The standard graduate text.

26.15 External links• Hazewinkel, Michiel, ed. (2001), “Manifold”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Dimensions-math.org (A film explaining and visualizing manifolds up to fourth dimension.)

• The manifold atlas project of the Max Planck Institute for Mathematics in Bonn

26.15. EXTERNAL LINKS 105

The chart maps the part of the sphere with positive z coordinate to a disc.

106 CHAPTER 26. MANIFOLD

A finite cylinder is a manifold with boundary.

26.15. EXTERNAL LINKS 107

Möbius strip

108 CHAPTER 26. MANIFOLD

The Klein bottle immersed in three-dimensional space

26.15. EXTERNAL LINKS 109

A Morin surface, an immersion used in sphere eversion

110 CHAPTER 26. MANIFOLD

3D color plot of the spherical harmonics of degree n = 5

Chapter 27

Mathematical analysis

A strange attractor arising from a differential equation. Differential equations are an important area of mathematical analysis withmany applications to science and engineering.

Mathematical analysis is a branch of mathematics that includes the theories of differentiation, integration, measure,limits, infinite series, and analytic functions.[1]

These theories are usually studied in the context of real and complex numbers and functions. Analysis evolvedfrom calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguishedfrom geometry; however, it can be applied to any space of mathematical objects that has a definition of nearness (atopological space) or specific distances between objects (a metric space).

111

112 CHAPTER 27. MATHEMATICAL ANALYSIS

27.1 History

Archimedes used the method of exhaustion to compute the area inside a circle by finding the area of regular polygons with more andmore sides. This was an early but informal example of a limit, one of the most basic concepts in mathematical analysis.

Mathematical analysis formally developed in the 17th century during the Scientific Revolution,[2] but many of its ideascan be traced back to earlier mathematicians. Early results in analysis were implicitly present in the early days ofancient Greek mathematics. For instance, an infinite geometric sum is implicit in Zeno’s paradox of the dichotomy.[3]Later, Greek mathematicians such as Eudoxus and Archimedes made more explicit, but informal, use of the conceptsof limits and convergence when they used the method of exhaustion to compute the area and volume of regions andsolids.[4] The explicit use of infinitesimals appears in Archimedes’ The Method of Mechanical Theorems, a workrediscovered in the 20th century.[5] In Asia, the Chinese mathematician Liu Hui used the method of exhaustionin the 3rd century AD to find the area of a circle.[6] Zu Chongzhi established a method that would later be calledCavalieri’s principle to find the volume of a sphere in the 5th century.[7] The Indian mathematician Bhāskara II gaveexamples of the derivative and used what is now known as Rolle’s theorem in the 12th century.[8]

In the 14th century, Madhava of Sangamagrama developed infinite series expansions, like the power series and theTaylor series, of functions such as sine, cosine, tangent and arctangent.[9] Alongside his development of the Taylorseries of the trigonometric functions, he also estimated the magnitude of the error terms created by truncating theseseries and gave a rational approximation of an infinite series. His followers at the Kerala school of astronomy andmathematics further expanded his works, up to the 16th century.The modern foundations of mathematical analysis were established in 17th century Europe.[2] Descartes and Fermatindependently developed analytic geometry, and a few decades later Newton and Leibniz independently developedinfinitesimal calculus, which grew, with the stimulus of applied work that continued through the 18th century, intoanalysis topics such as the calculus of variations, ordinary and partial differential equations, Fourier analysis, andgenerating functions. During this period, calculus techniques were applied to approximate discrete problems bycontinuous ones.In the 18th century, Euler introduced the notion of mathematical function.[10] Real analysis began to emerge as anindependent subject when Bernard Bolzano introduced the modern definition of continuity in 1816,[11] but Bolzano’swork did not becomewidely known until the 1870s. In 1821, Cauchy began to put calculus on a firm logical foundationby rejecting the principle of the generality of algebra widely used in earlier work, particularly by Euler. Instead,Cauchy formulated calculus in terms of geometric ideas and infinitesimals. Thus, his definition of continuity requiredan infinitesimal change in x to correspond to an infinitesimal change in y. He also introduced the concept of theCauchy sequence, and started the formal theory of complex analysis. Poisson, Liouville, Fourier and others studiedpartial differential equations and harmonic analysis. The contributions of these mathematicians and others, suchas Weierstrass, developed the (ε, δ)-definition of limit approach, thus founding the modern field of mathematicalanalysis.In the middle of the 19th century Riemann introduced his theory of integration. The last third of the century sawthe arithmetization of analysis by Weierstrass, who thought that geometric reasoning was inherently misleading, andintroduced the “epsilon-delta” definition of limit. Then, mathematicians started worrying that they were assuming theexistence of a continuum of real numbers without proof. Dedekind then constructed the real numbers by Dedekindcuts, in which irrational numbers are formally defined, which serve to fill the “gaps” between rational numbers, therebycreating a complete set: the continuum of real numbers, which had already been developed by Simon Stevin in termsof decimal expansions. Around that time, the attempts to refine the theorems of Riemann integration led to the study

27.2. IMPORTANT CONCEPTS 113

of the “size” of the set of discontinuities of real functions.Also, "monsters" (nowhere continuous functions, continuous but nowhere differentiable functions, space-filling curves)began to be investigated. In this context, Jordan developed his theory of measure, Cantor developed what is nowcalled naive set theory, and Baire proved the Baire category theorem. In the early 20th century, calculus was formal-ized using an axiomatic set theory. Lebesgue solved the problem of measure, and Hilbert introduced Hilbert spaces tosolve integral equations. The idea of normed vector space was in the air, and in the 1920s Banach created functionalanalysis.

27.2 Important concepts

27.2.1 Metric spaces

Main article: Metric space

In mathematics, a metric space is a set where a notion of distance (called a metric) between elements of the set isdefined.Much of analysis happens in some metric space; the most commonly used are the real line, the complex plane,Euclidean space, other vector spaces, and the integers. Examples of analysis without a metric include measure theory(which describes size rather than distance) and functional analysis (which studies topological vector spaces that neednot have any sense of distance).Formally, Ametric space is an ordered pair (M,d) whereM is a set and d is a metric onM , i.e., a function

d : M ×M → R

such that for any x, y, z ∈M , the following holds:

1. d(x, y) = 0 if and only if x = y (identity of indiscernibles),

2. d(x, y) = d(y, x) (symmetry) and

3. d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality) .

By taking the third property and letting z = x , it can be shown that d(x, y) ≥ 0 (non-negative).

27.2.2 Sequences and limits

Main article: Sequence

A sequence is an ordered list. Like a set, it contains members (also called elements, or terms). Unlike a set, ordermatters, and exactly the same elements can appear multiple times at different positions in the sequence. Most pre-cisely, a sequence can be defined as a function whose domain is a countable totally ordered set, such as the naturalnumbers.One of the most important properties of a sequence is convergence. Informally, a sequence converges if it has a limit.Continuing informally, a (singly-infinite) sequence has a limit if it approaches some point x, called the limit, as nbecomes very large. That is, for an abstract sequence (an) (with n running from 1 to infinity understood) the distancebetween an and x approaches 0 as n→∞, denoted

limn→∞

an = x.

114 CHAPTER 27. MATHEMATICAL ANALYSIS

27.3 Main branches

27.3.1 Real analysis

Main article: Real analysis

Real analysis (traditionally, the theory of functions of a real variable) is a branch of mathematical analysis dealingwith the real numbers and real-valued functions of a real variable.[12][13] In particular, it deals with the analyticproperties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculusof the real numbers, and continuity, smoothness and related properties of real-valued functions.

27.3.2 Complex analysis

Main article: Complex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of math-ematical analysis that investigates functions of complex numbers.[14] It is useful in many branches of mathematics,including algebraic geometry, number theory, applied mathematics; as well as in physics, including hydrodynamics,thermodynamics, mechanical engineering, electrical engineering, and particularly, quantum field theory.Complex analysis is particularly concerned with the analytic functions of complex variables (or, more generally,meromorphic functions). Because the separate real and imaginary parts of any analytic functionmust satisfy Laplace’sequation, complex analysis is widely applicable to two-dimensional problems in physics.

27.3.3 Functional analysis

Main article: Functional analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spacesendowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear operatorsacting upon these spaces and respecting these structures in a suitable sense.[15][16] The historical roots of functionalanalysis lie in the study of spaces of functions and the formulation of properties of transformations of functions suchas the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. Thispoint of view turned out to be particularly useful for the study of differential and integral equations.

27.3.4 Differential equations

Main article: Differential equations

A differential equation is a mathematical equation for an unknown function of one or several variables that relatesthe values of the function itself and its derivatives of various orders.[17][18][19] Differential equations play a prominentrole in engineering, physics, economics, biology, and other disciplines.Differential equations arise in many areas of science and technology, specifically whenever a deterministic relationinvolving some continuously varying quantities (modeled by functions) and their rates of change in space and/or time(expressed as derivatives) is known or postulated. This is illustrated in classical mechanics, where the motion of abody is described by its position and velocity as the time value varies. Newton’s laws allow one (given the position,velocity, acceleration and various forces acting on the body) to express these variables dynamically as a differentialequation for the unknown position of the body as a function of time. In some cases, this differential equation (calledan equation of motion) may be solved explicitly.

27.4. OTHER TOPICS IN MATHEMATICAL ANALYSIS 115

27.3.5 Measure theory

Main article: Measure (mathematics)

A measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpretedas its size.[20] In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularlyimportant example is the Lebesgue measure on a Euclidean space, which assigns the conventional length, area, andvolume of Euclidean geometry to suitable subsets of the n -dimensional Euclidean space Rn . For instance, theLebesguemeasure of the interval [0, 1] in the real numbers is its length in the everyday sense of the word – specifically,1.Technically, a measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a setX . Itmust assign 0 to the empty set and be (countably) additive: the measure of a 'large' subset that can be decomposedinto a finite (or countable) number of 'smaller' disjoint subsets, is the sum of the measures of the “smaller” subsets.In general, if one wants to associate a consistent size to each subset of a given set while satisfying the other axioms ofa measure, one only finds trivial examples like the counting measure. This problem was resolved by defining measureonly on a sub-collection of all subsets; the so-calledmeasurable subsets, which are required to form a σ -algebra. Thismeans that countable unions, countable intersections and complements of measurable subsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarilycomplicated in the sense of being badly mixed up with their complement. Indeed, their existence is a non-trivialconsequence of the axiom of choice.

27.3.6 Numerical analysis

Main article: Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolicmanipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics).[21]

Modern numerical analysis does not seek exact answers, because exact answers are often impossible to obtain inpractice. Instead, much of numerical analysis is concerned with obtaining approximate solutions while maintainingreasonable bounds on errors.Numerical analysis naturally finds applications in all fields of engineering and the physical sciences, but in the 21stcentury, the life sciences and even the arts have adopted elements of scientific computations. Ordinary differentialequations appear in celestial mechanics (planets, stars and galaxies); numerical linear algebra is important for dataanalysis; stochastic differential equations and Markov chains are essential in simulating living cells for medicine andbiology.

27.4 Other topics in mathematical analysis

• Calculus of variations deals with extremizing functionals, as opposed to ordinary calculus which deals withfunctions.

• Harmonic analysis deals with Fourier series and their abstractions.

• Geometric analysis involves the use of geometrical methods in the study of partial differential equations andthe application of the theory of partial differential equations to geometry.

• Clifford analysis, the study of Clifford valued functions that are annihilated by Dirac or Dirac-like operators,termed in general as monogenic or Clifford analytic functions.

• p-adic analysis, the study of analysis within the context of p-adic numbers, which differs in some interestingand surprising ways from its real and complex counterparts.

• Non-standard analysis, which investigates the hyperreal numbers and their functions and gives a rigorous treat-ment of infinitesimals and infinitely large numbers.

• Computable analysis, the study of which parts of analysis can be carried out in a computable manner.

116 CHAPTER 27. MATHEMATICAL ANALYSIS

• Stochastic calculus – analytical notions developed for stochastic processes.

• Set-valued analysis – applies ideas from analysis and topology to set-valued functions.

• Convex analysis, the study of convex sets and functions.

• Tropical analysis (or idempotent analysis) – analysis in the context of the semiring of the max-plus algebrawhere the lack of an additive inverse is compensated somewhat by the idempotent rule A + A = A. Whentransferred to the tropical setting, many nonlinear problems become linear.[22]

27.5 Applications

Techniques from analysis are also found in other areas such as:

27.5.1 Physical sciences

The vastmajority of classicalmechanics, relativity, and quantummechanics is based on applied analysis, and differentialequations in particular. Examples of important differential equations include Newton’s second law, the Schrödingerequation, and the Einstein field equations.Functional analysis is also a major factor in quantum mechanics.

27.5.2 Signal processing

When processing signals, such as audio, radio waves, light waves, seismic waves, and even images, Fourier analysiscan isolate individual components of a compound waveform, concentrating them for easier detection and/or removal.A large family of signal processing techniques consist of Fourier-transforming a signal, manipulating the Fourier-transformed data in a simple way, and reversing the transformation.[23]

27.5.3 Other areas of mathematics

Techniques from analysis are used in many areas of mathematics, including:

• Analytic number theory

• Analytic combinatorics

• Continuous probability

• Differential entropy in information theory

• Differential games

• Differential geometry, the application of calculus to specific mathematical spaces known as manifolds thatpossess a complicated internal structure but behave in a simple manner locally.

• Differential topology

• Mathematical finance

27.6 See also

• Constructive analysis

• History of calculus

• Non-classical analysis

27.7. NOTES 117

• Paraconsistent mathematics

• Smooth infinitesimal analysis

• Timeline of calculus and mathematical analysis

27.7 Notes[1] Edwin Hewitt and Karl Stromberg, “Real and Abstract Analysis”, Springer-Verlag, 1965

[2] Jahnke, Hans Niels (2003). A History of Analysis. American Mathematical Society. p. 7. ISBN 978-0-8218-2623-2.

[3] Stillwell (2004). “Infinite Series”. Mathematics and its History (2nd ed.). Springer Science + Business Media Inc. p. 170.ISBN 0-387-95336-1. Infinite series were present in Greek mathematics, [...] There is no question that Zeno’s paradox ofthe dichotomy (Section 4.1), for example, concerns the decomposition of the number 1 into the infinite series 1 ⁄2 + 1 ⁄22+ 1 ⁄23 + 1 ⁄24 + ... and that Archimedes found the area of the parabolic segment (Section 4.4) essentially by summing theinfinite series 1 + 1 ⁄4 + 1 ⁄42 + 1 ⁄43 + ... = 4 ⁄3. Both these examples are special cases of the result we express as summationof a geometric series

[4] (Smith, 1958)

[5] Pinto, J. Sousa (2004). Infinitesimal Methods of Mathematical Analysis. Horwood Publishing. p. 8. ISBN 978-1-898563-99-0.

[6] Dun, Liu; Fan, Dainian; Cohen, Robert Sonné (1966). “A comparison of Archimdes’ and Liu Hui’s studies of circles”.Chinese studies in the history and philosophy of science and technology 130. Springer. p. 279. ISBN 0-7923-3463-9.,Chapter , p. 279

[7] Zill, Dennis G.; Wright, Scott; Wright, Warren S. (2009). Calculus: Early Transcendentals (3 ed.). Jones & BartlettLearning. p. xxvii. ISBN 0-7637-5995-3., Extract of page 27

[8] Seal, Sir Brajendranath (1915), The positive sciences of the ancient Hindus, Longmans, Green and co.

[9] C. T. Rajagopal and M. S. Rangachari (June 1978). “On an untapped source of medieval Keralese Mathematics”. Archivefor History of Exact Sciences 18 (2): 89–102. doi:10.1007/BF00348142.

[10] Dunham, William (1999). Euler: The Master of Us All. The Mathematical Association of America. p. 17.

[11] • Cooke, Roger (1997). “Beyond the Calculus”. The History of Mathematics: A Brief Course. Wiley-Interscience. p.379. ISBN 0-471-18082-3. Real analysis began its growth as an independent subject with the introduction of themodern definition of continuity in 1816 by the Czech mathematician Bernard Bolzano (1781–1848)

[12] Rudin, Walter. Principles of Mathematical Analysis. Walter Rudin Student Series in Advanced Mathematics (3rd ed.).McGraw–Hill. ISBN 978-0-07-054235-8.

[13] Abbott, Stephen (2001). Understanding Analysis. Undergradutate Texts in Mathematics. New York: Springer-Verlag.ISBN 0-387-95060-5.

[14] Ahlfors.,Complex Analysis (McGraw-Hill)

[15] Rudin, W.: Functional Analysis, McGraw-Hill Science, 1991

[16] Conway, J. B.: A Course in Functional Analysis, 2nd edition, Springer-Verlag, 1994, ISBN 0-387-97245-5

[17] E. L. Ince, Ordinary Differential Equations, Dover Publications, 1958, ISBN 0-486-60349-0

[18] Witold Hurewicz, Lectures on Ordinary Differential Equations, Dover Publications, ISBN 0-486-49510-8

[19] Evans, L. C. (1998), Partial Differential Equations, Providence: American Mathematical Society, ISBN 0-8218-0772-2

[20] Terence Tao, 2011. An Introduction to Measure Theory. American Mathematical Society.

[21] Hildebrand, F. B. (1974). Introduction to Numerical Analysis (2nd edition ed.). McGraw-Hill. ISBN 0-07-028761-9.

[22] THE MASLOV DEQUANTIZATION, IDEMPOTENT AND TROPICAL MATHEMATICS: A BRIEF INTRODUC-TION

[23] Theory and application of digital signal processing Rabiner, L. R.; Gold, B. Englewood Cliffs, N.J., Prentice-Hall, Inc.,1975.

118 CHAPTER 27. MATHEMATICAL ANALYSIS

27.8 References• Aleksandrov, A. D., Kolmogorov, A. N., Lavrent'ev, M. A. (eds.). 1984. Mathematics, its Content, Methods,and Meaning. 2nd ed. Translated by S. H. Gould, K. A. Hirsch and T. Bartha; translation edited by S. H.Gould. MIT Press; published in cooperation with the American Mathematical Society.

• Apostol, Tom M. 1974. Mathematical Analysis. 2nd ed. Addison–Wesley. ISBN 978-0-201-00288-1.

• Binmore, K.G. 1980–1981. The foundations of analysis: a straightforward introduction. 2 volumes. Cam-bridge University Press.

• Johnsonbaugh, Richard, & W. E. Pfaffenberger. 1981. Foundations of mathematical analysis. New York: M.Dekker.

• Nikol’skii, S. M. 2002. “Mathematical analysis”. In Encyclopaedia of Mathematics, Michiel Hazewinkel (edi-tor). Springer-Verlag. ISBN 1-4020-0609-8.

• Rombaldi, Jean-Étienne. 2004. Éléments d'analyse réelle : CAPES et agrégation interne de mathématiques.EDP Sciences. ISBN 2-86883-681-X.

• Rudin, Walter. 1976. Principles of Mathematical Analysis. McGraw–Hill Publishing Co.; 3rd revised edition(September 1, 1976), ISBN 978-0-07-085613-4.

• Smith, David E. 1958. History of Mathematics. Dover Publications. ISBN 0-486-20430-8.

• Whittaker, E. T. and Watson, G. N.. 1927. A Course of Modern Analysis. 4th edition. Cambridge UniversityPress. ISBN 0-521-58807-3.

• Real Analysis - Course Notes

27.9 External links• Earliest Known Uses of Some of the Words of Mathematics: Calculus & Analysis

• Basic Analysis: Introduction to Real Analysis by Jiri Lebl (Creative Commons BY-NC-SA)

Chapter 28

Mesocompact space

In mathematics, in the field of general topology, a topological space is said to bemesocompact if every open cover hasa compact-finite open refinement.[1] That is, given any open cover, we can find an open refinement with the propertythat every compact set meets only finitely many members of the refinement.[2]

The following facts are true about mesocompactness:

• Every compact space, and more generally every paracompact space is mesocompact. This follows from thefact that any locally finite cover is automatically compact-finite.

• Every mesocompact space is metacompact, and hence also orthocompact. This follows from the fact that pointsare compact, and hence any compact-finite cover is automatically point finite.

28.1 Notes[1] Hart, Nagata & Vaughan, p200

[2] Pearl, p23

28.2 References• K.P. Hart; J. Nagata; J.E. Vaughan, eds. (2004), Encyclopedia of General Topology, Elsevier, ISBN 0-444-50355-2

• Pearl, Elliott, ed. (2007), Open Problems in Topology II, Elsevier, ISBN 0-444-52208-5

119

Chapter 29

Metacompact space

In mathematics, in the field of general topology, a topological space is said to be metacompact if every open coverhas a point finite open refinement. That is, given any open cover of the topological space, there is a refinement whichis again an open cover with the property that every point is contained only in finitely many sets of the refining cover.A space is countably metacompact if every countable open cover has a point finite open refinement.

29.1 Properties

The following can be said about metacompactness in relation to other properties of topological spaces:

• Every paracompact space is metacompact. This implies that every compact space is metacompact, and everymetric space is metacompact. The converse does not hold: a counter-example is the Dieudonné plank.

• Every metacompact space is orthocompact.

• Every metacompact normal space is a shrinking space

• The product of a compact space and a metacompact space is metacompact. This follows from the tube lemma.

• An easy example of a non-metacompact space (but a countably metacompact space) is the Moore plane.

• In order for a Tychonoff space X to be compact it is necessary and sufficient that X be metacompact andpseudocompact (see Watson).

29.2 Covering dimension

A topological space X is said to be of covering dimension n if every open cover of X has a point finite open refinementsuch that no point of X is included in more than n + 1 sets in the refinement and if n is the minimum value for whichthis is true. If no such minimal n exists, the space is said to be of infinite covering dimension.

29.3 See also

• Compact space

• Paracompact space

• Normal space

• Realcompact space

• Pseudocompact space

120

29.4. REFERENCES 121

• Mesocompact space

• Tychonoff space

• Glossary of topology

29.4 References• Watson, W. Stephen (1981). “Pseudocompact metacompact spaces are compact”. Proc. Amer. Math. Soc.81: 151–152. doi:10.1090/s0002-9939-1981-0589159-1.

• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978]. Counterexamples in Topology (Dover reprint of1978 ed.). Berlin, New York: Springer-Verlag. ISBN 978-0-486-68735-3. MR 507446. P.23.

Chapter 30

Metric space

In mathematics, a metric space is a set for which distances between all members of the set are defined. Thosedistances, taken together, are called a metric on the set.The most familiar metric space is 3-dimensional Euclidean space. In fact, a “metric” is the generalization of theEuclidean metric arising from the four long-known properties of the Euclidean distance. The Euclidean metricdefines the distance between two points as the length of the straight line segment connecting them. Other metricspaces occur for example in elliptic geometry and hyperbolic geometry, where distance on a sphere measured byangle is a metric, and the hyperboloid model of hyperbolic geometry is used by special relativity as a metric space ofvelocities.A metric on a space induces topological properties like open and closed sets, which lead to the study of more abstracttopological spaces.In the most general definition of a metric space, the distance between set elements can be negative. Spaces like theseare important in the theory of relativity.

30.1 History

Maurice Fréchet introduced metric spaces in his work Sur quelques points du calcul fonctionnel, Rendic. Circ. Mat.Palermo 22 (1906) 1–74.

30.2 Definition

Ametric space is an ordered pair (M,d) whereM is a set and d is a metric onM , i.e., a function

d : M ×M → R

such that for any x, y, z ∈M , the following holds:[1]

1. d(x, y) ≥ 0 (non-negative),

2. d(x, y) = 0 ⇐⇒ x = y (identity of indiscernibles),

3. d(x, y) = d(y, x) (symmetry) and

4. d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality) .

The first condition follows from the other three, since: for any x, y ∈M ,

122

30.3. EXAMPLES OF METRIC SPACES 123

d(x, y) + d(y, x) ≥ d(x, x) inequality) triangle (by=⇒ d(x, y) + d(x, y) ≥ d(x, x) symmetry) (by=⇒ 2d(x, y) ≥ 0 indiscernibles) of identity (by=⇒ d(x, y) ≥ 0.

The function d is also called distance function or simply distance. Often, d is omitted and one just writes M for ametric space if it is clear from the context what metric is used.Ignoring mathematical details, for any system of roads and terrains the distance between two locations can be definedas the length of the shortest route connecting those locations. To be a metric there shouldn't be any one-way roads.The triangle inequality expresses the fact that detours aren't shortcuts. Many of the examples below can be seen asconcrete versions of this general idea.

30.3 Examples of metric spaces

• The real numbers with the distance function d(x, y) = |y − x| given by the absolute difference, and moregenerally Euclidean n -space with the Euclidean distance, are complete metric spaces. The rational numberswith the same distance also form a metric space, but are not complete.

• The positive real numbers with distance function d(x, y) = | log(y/x)| is a complete metric space.

• Any normed vector space is a metric space by defining d(x, y) = ∥y − x∥ , see also metrics on vector spaces.(If such a space is complete, we call it a Banach space.) Examples:

• The Manhattan norm gives rise to the Manhattan distance, where the distance between any two points,or vectors, is the sum of the differences between corresponding coordinates.

• The maximum norm gives rise to the Chebyshev distance or chessboard distance, the minimal numberof moves a chess king would take to travel from x to y .

• The British Rail metric (also called the Post Office metric or the SNCF metric) on a normed vector space isgiven by d(x, y) = ∥x∥ + ∥y∥ for distinct points x and y , and d(x, x) = 0 . More generally ∥.∥ can bereplaced with a function f taking an arbitrary set S to non-negative reals and taking the value 0 at most once:then the metric is defined on S by d(x, y) = f(x) + f(y) for distinct points x and y , and d(x, x) = 0 . Thename alludes to the tendency of railway journeys (or letters) to proceed via London (or Paris) irrespective oftheir final destination.

• If (M,d) is a metric space and X is a subset of M , then (X, d) becomes a metric space by restricting thedomain of d to X ×X .

• The discrete metric, where d(x, y) = 0 if x = y and d(x, y) = 1 otherwise, is a simple but important example,and can be applied to all non-empty sets. This, in particular, shows that for any non-empty set, there is alwaysa metric space associated to it. Using this metric, any point is an open ball, and therefore every subset is openand the space has the discrete topology.

• A finite metric space is a metric space having a finite number of points. Not every finite metric space can beisometrically embedded in a Euclidean space.[2][3]

• The hyperbolic plane is a metric space. More generally:

• IfM is any connected Riemannian manifold, then we can turnM into a metric space by defining the distanceof two points as the infimum of the lengths of the paths (continuously differentiable curves) connecting them.

• If X is some set and M is a metric space, then, the set of all bounded functions f : X → M (i.e. thosefunctions whose image is a bounded subset of M ) can be turned into a metric space by defining d(f, g) =supx∈X d(f(x), g(x)) for any two bounded functions f and g (where sup is supremum.[4] This metric is calledthe uniform metric or supremum metric, and IfM is complete, then this function space is complete as well.If X is also a topological space, then the set of all bounded continuous functions fromX toM (endowed withthe uniform metric), will also be a complete metric if M is.

124 CHAPTER 30. METRIC SPACE

• If G is an undirected connected graph, then the set V of vertices of G can be turned into a metric space bydefining d(x, y) to be the length of the shortest path connecting the vertices x and y . In geometric grouptheory this is applied to the Cayley graph of a group, yielding the word metric.

• The Levenshtein distance is a measure of the dissimilarity between two strings u and v , defined as the minimalnumber of character deletions, insertions, or substitutions required to transform u into v . This can be thoughtof as a special case of the shortest path metric in a graph and is one example of an edit distance.

• Given a metric space (X, d) and an increasing concave function f : [0,∞) → [0,∞) such that f(x) = 0 ifand only if x = 0 , then f d is also a metric on X .

• Given an injective function f from any set A to a metric space (X, d) , d(f(x), f(y)) defines a metric on A .

• Using T-theory, the tight span of a metric space is also a metric space. The tight span is useful in several typesof analysis.

• The set of allm by n matrices over some field is a metric space with respect to the rank distance d(X,Y ) =rank(Y −X) .

• The Helly metric is used in game theory.

30.4 Open and closed sets, topology and convergence

Every metric space is a topological space in a natural manner, and therefore all definitions and theorems about generaltopological spaces also apply to all metric spaces.About any point x in a metric spaceM we define the open ball of radius r > 0 (where r is a real number) aboutx as the set

B(x; r) = y ∈M : d(x, y) < r.

These open balls form the base for a topology on M, making it a topological space.Explicitly, a subset U ofM is called open if for every x in U there exists an r > 0 such that B(x; r) is contained inU . The complement of an open set is called closed. A neighborhood of the point x is any subset ofM that containsan open ball about x as a subset.A topological space which can arise in this way from a metric space is called a metrizable space; see the article onmetrization theorems for further details.A sequence ( xn ) in a metric spaceM is said to converge to the limit x ∈M iff for every ϵ > 0 , there exists a naturalnumber N such that d(xn, x) < ϵ for all n > N . Equivalently, one can use the general definition of convergenceavailable in all topological spaces.A subset A of the metric spaceM is closed iff every sequence in A that converges to a limit inM has its limit in A .

30.5 Types of metric spaces

30.5.1 Complete spaces

Main article: Complete metric space

Ametric spaceM is said to be complete if every Cauchy sequence converges inM . That is to say: if d(xn, xm) → 0as both n andm independently go to infinity, then there is some y ∈M with d(xn, y) → 0 .Every Euclidean space is complete, as is every closed subset of a complete space. The rational numbers, using theabsolute value metric d(x, y) = |x− y| , are not complete.Every metric space has a unique (up to isometry) completion, which is a complete space that contains the given spaceas a dense subset. For example, the real numbers are the completion of the rationals.

30.5. TYPES OF METRIC SPACES 125

IfX is a complete subset of the metric spaceM , thenX is closed inM . Indeed, a space is complete iff it is closedin any containing metric space.Every complete metric space is a Baire space.

30.5.2 Bounded and totally bounded spaces

A

diam(A)

Diameter of a set.

See also: bounded set

A metric space M is called bounded if there exists some number r, such that d(x,y) ≤ r for all x and y in M. Thesmallest possible such r is called the diameter of M. The space M is called precompact or totally bounded if forevery r > 0 there exist finitely many open balls of radius r whose union coversM. Since the set of the centres of these

126 CHAPTER 30. METRIC SPACE

balls is finite, it has finite diameter, from which it follows (using the triangle inequality) that every totally boundedspace is bounded. The converse does not hold, since any infinite set can be given the discrete metric (one of theexamples above) under which it is bounded and yet not totally bounded.Note that in the context of intervals in the space of real numbers and occasionally regions in a Euclidean space Rn

a bounded set is referred to as “a finite interval” or “finite region”. However boundedness should not in general beconfused with “finite”, which refers to the number of elements, not to how far the set extends; finiteness impliesboundedness, but not conversely. Also note that an unbounded subset of Rn may have a finite volume.

30.5.3 Compact spaces

Ametric spaceM is compact if every sequence inM has a subsequence that converges to a point inM. This is knownas sequential compactness and, in metric spaces (but not in general topological spaces), is equivalent to the topologicalnotions of countable compactness and compactness defined via open covers.Examples of compact metric spaces include the closed interval [0,1] with the absolute value metric, all metric spaceswith finitely many points, and the Cantor set. Every closed subset of a compact space is itself compact.A metric space is compact iff it is complete and totally bounded. This is known as the Heine–Borel theorem. Notethat compactness depends only on the topology, while boundedness depends on the metric.Lebesgue’s number lemma states that for every open cover of a compact metric space M, there exists a “Lebesguenumber” δ such that every subset of M of diameter < δ is contained in some member of the cover.Every compact metric space is second countable,[5] and is a continuous image of the Cantor set. (The latter result isdue to Pavel Alexandrov and Urysohn.)

30.5.4 Locally compact and proper spaces

A metric space is said to be locally compact if every point has a compact neighborhood. Euclidean spaces are locallycompact, but infinite-dimensional Banach spaces are not.A space is proper if every closed ball y : d(x,y) ≤ r is compact. Proper spaces are locally compact, but the converseis not true in general.

30.5.5 Connectedness

A metric spaceM is connected if the only subsets that are both open and closed are the empty set andM itself.A metric spaceM is path connected if for any two points x, y ∈ M there exists a continuous map f : [0, 1] → Mwith f(0) = x and f(1) = y . Every path connected space is connected, but the converse is not true in general.There are also local versions of these definitions: locally connected spaces and locally path connected spaces.Simply connected spaces are those that, in a certain sense, do not have “holes”.

30.5.6 Separable spaces

A metric space is separable space if it has a countable dense subset. Typical examples are the real numbers orany Euclidean space. For metric spaces (but not for general topological spaces) separability is equivalent to secondcountability and also to the Lindelöf property.

30.6 Types of maps between metric spaces

Suppose (M1,d1) and (M2,d2) are two metric spaces.

30.6. TYPES OF MAPS BETWEEN METRIC SPACES 127

30.6.1 Continuous maps

Main article: Continuous function (topology)

The map f:M1→M2 is continuous if it has one (and therefore all) of the following equivalent properties:

General topological continuity for every open set U in M2, the preimage f −1(U) is open in M1

This is the general definition of continuity in topology.

Sequential continuity if (xn) is a sequence in M1 that converges to x in M1, then the sequence (f(xn)) convergesto f(x) in M2.

This is sequential continuity, due to Eduard Heine.

ε-δ definition for every x in M1 and every ε>0 there exists δ>0 such that for all y in M1 we have

d1(x, y) < δ ⇒ d2(f(x), f(y)) < ε.

This uses the (ε, δ)-definition of limit, and is due to Augustin Louis Cauchy.

Moreover, f is continuous if and only if it is continuous on every compact subset of M1.The image of every compact set under a continuous function is compact, and the image of every connected set undera continuous function is connected.

30.6.2 Uniformly continuous maps

The map ƒ : M1 → M2 is uniformly continuous if for every ε > 0 there exists δ > 0 such that

d1(x, y) < δ ⇒ d2(f(x), f(y)) < ε for all x, y ∈M1.

Every uniformly continuous map ƒ : M1 →M2 is continuous. The converse is true ifM1 is compact (Heine–Cantortheorem).Uniformly continuous maps turn Cauchy sequences in M1 into Cauchy sequences in M2. For continuous maps thisis generally wrong; for example, a continuous map from the open interval (0,1) onto the real line turns some Cauchysequences into unbounded sequences.

30.6.3 Lipschitz-continuous maps and contractions

Given a number K > 0, the map ƒ : M1 → M2 is K-Lipschitz continuous if

d2(f(x), f(y)) ≤ Kd1(x, y) for all x, y ∈M1.

Every Lipschitz-continuous map is uniformly continuous, but the converse is not true in general.If K < 1, then ƒ is called a contraction. SupposeM2 =M1 andM1 is complete. If ƒ is a contraction, then ƒ admits aunique fixed point (Banach fixed point theorem). If M1 is compact, the condition can be weakened a bit: ƒ admits aunique fixed point if

d(f(x), f(y)) < d(x, y) for all x = y ∈M1

128 CHAPTER 30. METRIC SPACE

30.6.4 Isometries

The map f:M1→M2 is an isometry if

d2(f(x), f(y)) = d1(x, y) for all x, y ∈M1

Isometries are always injective; the image of a compact or complete set under an isometry is compact or complete,respectively. However, if the isometry is not surjective, then the image of a closed (or open) set need not be closed(or open).

30.6.5 Quasi-isometries

The map f : M1 → M2 is a quasi-isometry if there exist constants A ≥ 1 and B ≥ 0 such that

1

Ad2(f(x), f(y))−B ≤ d1(x, y) ≤ Ad2(f(x), f(y)) +B all for x, y ∈M1

and a constant C ≥ 0 such that every point in M2 has a distance at most C from some point in the image f(M1).Note that a quasi-isometry is not required to be continuous. Quasi-isometries compare the “large-scale structure” ofmetric spaces; they find use in geometric group theory in relation to the word metric.

30.7 Notions of metric space equivalence

Given two metric spaces (M1, d1) and (M2, d2):

• They are called homeomorphic (topologically isomorphic) if there exists a homeomorphism between them(i.e., a bijection continuous in both directions).

• They are called uniformic (uniformly isomorphic) if there exists a uniform isomorphism between them (i.e.,a bijection uniformly continuous in both directions).

• They are called isometric if there exists a bijective isometry between them. In this case, the two metric spacesare essentially identical.

• They are called quasi-isometric if there exists a quasi-isometry between them.

30.8 Topological properties

Metric spaces are paracompact[6] Hausdorff spaces[7] and hence normal (indeed they are perfectly normal). Animportant consequence is that every metric space admits partitions of unity and that every continuous real-valuedfunction defined on a closed subset of a metric space can be extended to a continuous map on the whole space (Tietzeextension theorem). It is also true that every real-valued Lipschitz-continuous map defined on a subset of a metricspace can be extended to a Lipschitz-continuous map on the whole space.Metric spaces are first countable since one can use balls with rational radius as a neighborhood base.The metric topology on a metric spaceM is the coarsest topology onM relative to which the metric d is a continuousmap from the product of M with itself to the non-negative real numbers.

30.9. DISTANCE BETWEEN POINTS AND SETS; HAUSDORFF DISTANCE AND GROMOV METRIC 129

30.9 Distance between points and sets; Hausdorff distance and Gromovmetric

A simple way to construct a function separating a point from a closed set (as required for a completely regular space)is to consider the distance between the point and the set. If (M,d) is a metric space, S is a subset ofM and x is a pointof M, we define the distance from x to S as

d(x, S) = infd(x, s) : s ∈ S where inf represents the infimum.

Then d(x, S) = 0 if and only if x belongs to the closure of S. Furthermore, we have the following generalization of thetriangle inequality:

d(x, S) ≤ d(x, y) + d(y, S),

which in particular shows that the map x 7→ d(x, S) is continuous.Given two subsets S and T of M, we define their Hausdorff distance to be

dH(S, T ) = maxsupd(s, T ) : s ∈ S, supd(t, S) : t ∈ T where sup represents the supremum.

In general, the Hausdorff distance dH(S,T) can be infinite. Two sets are close to each other in the Hausdorff distanceif every element of either set is close to some element of the other set.The Hausdorff distance dH turns the set K(M) of all non-empty compact subsets of M into a metric space. One canshow that K(M) is complete ifM is complete. (A different notion of convergence of compact subsets is given by theKuratowski convergence.)One can then define the Gromov–Hausdorff distance between any two metric spaces by considering the minimalHausdorff distance of isometrically embedded versions of the two spaces. Using this distance, the class of all (isometryclasses of) compact metric spaces becomes a metric space in its own right.

30.10 Product metric spaces

If (M1, d1), . . . , (Mn, dn) aremetric spaces, andN is the Euclidean norm onRn, then(M1×. . .×Mn, N(d1, . . . , dn)

)is a metric space, where the product metric is defined by

N(d1, ..., dn)((x1, . . . , xn), (y1, . . . , yn)

)= N

(d1(x1, y1), . . . , dn(xn, yn)

),

and the induced topology agrees with the product topology. By the equivalence of norms in finite dimensions, anequivalent metric is obtained if N is the taxicab norm, a p-norm, the max norm, or any other norm which is non-decreasing as the coordinates of a positive n-tuple increase (yielding the triangle inequality).Similarly, a countable product of metric spaces can be obtained using the following metric

d(x, y) =

∞∑i=1

1

2idi(xi, yi)

1 + di(xi, yi).

An uncountable product of metric spaces need not be metrizable. For example, RR is not first-countable and thusisn't metrizable.

30.10.1 Continuity of distance

It is worth noting that in the case of a single space (M,d) , the distance map d : M ×M → R+ (from the definition)is uniformly continuous with respect to any of the above product metrics N(d, d) , and in particular is continuouswith respect to the product topology ofM ×M .

130 CHAPTER 30. METRIC SPACE

30.11 Quotient metric spaces

If M is a metric space with metric d, and ~ is an equivalence relation on M, then we can endow the quotient set M/~with the following (pseudo)metric. Given two equivalence classes [x] and [y], we define

d′([x], [y]) = infd(p1, q1) + d(p2, q2) + · · ·+ d(pn, qn)

where the infimum is taken over all finite sequences (p1, p2, . . . , pn) and (q1, q2, . . . , qn) with [p1] = [x] , [qn] = [y], [qi] = [pi+1], i = 1, 2, . . . , n − 1 . In general this will only define a pseudometric, i.e. d′([x], [y]) = 0 doesnot necessarily imply that [x] = [y] . However, for nice equivalence relations (e.g., those given by gluing togetherpolyhedra along faces), it is a metric. Moreover, if M is a compact space, then the induced topology on M/~ is thequotient topology.The quotient metric d is characterized by the following universal property. If f : (M,d) −→ (X, δ) is a metric mapbetween metric spaces (that is, δ(f(x), f(y)) ≤ d(x, y) for all x, y) satisfying f(x)=f(y) whenever x ∼ y, then theinduced function f : M/ ∼−→ X , given by f([x]) = f(x) , is a metric map f : (M/ ∼, d′) −→ (X, δ).

A topological space is sequential if and only if it is a quotient of a metric space.[8]

30.12 Generalizations of metric spaces• Every metric space is a uniform space in a natural manner, and every uniform space is naturally a topologicalspace. Uniform and topological spaces can therefore be regarded as generalizations of metric spaces.

• If we consider the first definition of a metric space given above and relax the second requirement, we arriveat the concepts of a pseudometric space or a dislocated metric space.[9] If we remove the third or fourth, wearrive at a quasimetric space, or a semimetric space.

• If the distance function takes values in the extended real number line R∪+∞, but otherwise satisfies all fourconditions, then it is called an extended metric and the corresponding space is called an ∞ -metric space. If thedistance function takes values in some (suitable) ordered set (and the triangle inequality is adjusted accordingly),then we arrive at the notion of generalized ultrametric.[10]

• Approach spaces are a generalization ofmetric spaces, based on point-to-set distances, instead of point-to-pointdistances.

• A continuity space is a generalization of metric spaces and posets, that can be used to unify the notions ofmetric spaces and domains.

• A partial metric space is intended to be the least generalisation of the notion of a metric space, such that thedistance of each point from itself is no longer necessarily zero.[11]

30.12.1 Metric spaces as enriched categories

The ordered set (R,≥) can be seen as a category by requesting exactly one morphism a → b if a ≥ b and noneotherwise. By using+ as the tensor product and 0 as the identity, it becomes a monoidal category R∗ . Every metricspace (M,d) can now be viewed as a categoryM∗ enriched over R∗ :

• Set Ob(M∗) :=M

• For each X,Y ∈M set Hom(X,Y ) := d(X,Y ) ∈ Ob(R∗)

• The composition morphism Hom(Y, Z) ⊗ Hom(X,Y ) → Hom(X,Z) will be the unique morphism in R∗

given from the triangle inequality d(y, z) + d(x, y) ≥ d(x, z)

• The identity morphism 0 → Hom(X,X) will be the unique morphism given from the fact that 0 ≥ d(X,X) .

30.13. SEE ALSO 131

• Since R∗ is a strict monoidal category, all diagrams that are required for an enriched category commute auto-matically.

See the paper by F.W. Lawvere listed below.

30.13 See also• Space (mathematics)

• Metric (mathematics)

• Metric signature

• Metric tensor

• Metric tree

• Norm (mathematics)

• Normed vector space

• Measure (mathematics)

• Hilbert space

• Product metric

• Aleksandrov–Rassias problem

• Category of metric spaces

• Classical Wiener space

• Glossary of Riemannian and metric geometry

• Isometry, contraction mapping and metric map

• Lipschitz continuity

• Triangle inequality

30.14 Notes[1] B. Choudhary (1992). The Elements of Complex Analysis. New Age International. p. 20. ISBN 978-81-224-0399-2.

[2] Nathan Linial. Finite Metric Spaces—Combinatorics, Geometry and Algorithms, Proceedings of the ICM, Beijing 2002,vol. 3, pp573–586

[3] Open problems on embeddings of finite metric spaces, edited by Jirīı Matoušek, 2007

[4] Searcóid, p. 107.

[5] PlanetMath: a compact metric space is second countable

[6] Rudin, Mary Ellen. A new proof that metric spaces are paracompact. Proceedings of the American Mathematical Society,Vol. 20, No. 2. (Feb., 1969), p. 603.

[7] metric spaces are Hausdorff at PlanetMath.org.

[8] Goreham, Anthony. Sequential convergence in Topological Spaces. Honours’ Dissertation, Queen’s College, Oxford (April,2001), p. 14

[9] Pascal Hitzler and Anthony Seda, Mathematical Aspects of Logic Programming Semantics. Chapman and Hall/CRC,2010.

[10] Pascal Hitzler and Anthony Seda, Mathematical Aspects of Logic Programming Semantics. Chapman and Hall/CRC,2010.

[11] http://www.dcs.warwick.ac.uk/pmetric/

132 CHAPTER 30. METRIC SPACE

30.15 References• Victor Bryant, Metric Spaces: Iteration and Application, Cambridge University Press, 1985, ISBN 0-521-31897-1.

• Dmitri Burago, Yu D Burago, Sergei Ivanov, A Course in Metric Geometry, American Mathematical Society,2001, ISBN 0-8218-2129-6.

• Athanase Papadopoulos,Metric Spaces, Convexity and Nonpositive Curvature, European Mathematical Society,First edition 2004, ISBN 978-3-03719-010-4. Second edition 2014, ISBN 978-3-03719-132-3.

• Mícheál Ó Searcóid,Metric Spaces, Springer Undergraduate Mathematics Series, 2006, ISBN 1-84628-369-8.

• Lawvere, F. William, “Metric spaces, generalized logic, and closed categories”, [Rend. Sem. Mat. Fis. Milano43 (1973), 135—166 (1974); (Italian summary)

This is reprinted (with author commentary) at Reprints in Theory and Applications of Categories Also (with an authorcommentary) in Enriched categories in the logic of geometry and analysis. Repr. Theory Appl. Categ. No. 1 (2002),1–37.

• Weisstein, Eric W., “Product Metric”, MathWorld.

30.16 External links• Hazewinkel, Michiel, ed. (2001), “Metric space”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Far and near — several examples of distance functions at cut-the-knot.

Chapter 31

Metrization theorem

In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to ametric space. That is, a topological space (X, τ) is said to be metrizable if there is a metric

d : X ×X → [0,∞)

such that the topology induced by d is τ . Metrization theorems are theorems that give sufficient conditions for atopological space to be metrizable.

31.1 Properties

Metrizable spaces inherit all topological properties frommetric spaces. For example, they are Hausdorff paracompactspaces (and hence normal and Tychonoff) and first-countable. However, some properties of the metric, such as com-pleteness, cannot be said to be inherited. This is also true of other structures linked to the metric. A metrizableuniform space, for example, may have a different set of contraction maps than a metric space to which it is homeo-morphic.

31.2 Metrization theorems

One of the first widely-recognized metrization theorems was Urysohn’s metrization theorem. This states thatevery Hausdorff second-countable regular space is metrizable. So, for example, every second-countable manifold ismetrizable. (Historical note: The form of the theorem shown here was in fact proved by Tychonoff in 1926. WhatUrysohn had shown, in a paper published posthumously in 1925, was that every second-countable normal Hausdorffspace is metrizable). The converse does not hold: there exist metric spaces that are not second countable, for example,an uncountable set endowed with the discrete metric.[1] The Nagata–Smirnov metrization theorem, described below,provides a more specific theorem where the converse does hold.Several other metrization theorems follow as simple corollaries to Urysohn’s Theorem. For example, a compactHausdorff space is metrizable if and only if it is second-countable.Urysohn’s Theorem can be restated as: A topological space is separable and metrizable if and only if it is regular,Hausdorff and second-countable. The Nagata–Smirnov metrization theorem extends this to the non-separable case.It states that a topological space is metrizable if and only if it is regular, Hausdorff and has a σ-locally finite base. Aσ-locally finite base is a base which is a union of countably many locally finite collections of open sets. For a closelyrelated theorem see the Bing metrization theorem.Separable metrizable spaces can also be characterized as those spaces which are homeomorphic to a subspace of theHilbert cube [0, 1]N , i.e. the countably infinite product of the unit interval (with its natural subspace topology fromthe reals) with itself, endowed with the product topology.A space is said to be locally metrizable if every point has a metrizable neighbourhood. Smirnov proved that a locallymetrizable space is metrizable if and only if it is Hausdorff and paracompact. In particular, a manifold is metrizable

133

134 CHAPTER 31. METRIZATION THEOREM

if and only if it is paracompact.

31.3 Examples

The group of unitary operators U(H) on a separable Hilbert space H endowed with the strong operator topology ismetrizable (see Proposition II.1 in [2]).

31.4 Examples of non-metrizable spaces

Non-normal spaces cannot be metrizable; important examples include

• the Zariski topology on an algebraic variety or on the spectrum of a ring, used in algebraic geometry,

• the topological vector space of all functions from the real line R to itself, with the topology of pointwiseconvergence.

The real line with the lower limit topology is not metrizable. The usual distance function is not a metric on thisspace because the topology it determines is the usual topology, not the lower limit topology. This space is Hausdorff,paracompact and first countable.The long line is locally metrizable but not metrizable; in a sense it is “too long”.

31.5 See also• Uniformizability, the property of a topological space of being homeomorphic to a uniform space, or equiva-lently the topology being defined by a family of pseudometrics

• Moore space (topology)

• Apollonian metric

• Nagata–Smirnov metrization theorem

• Bing metrization theorem

31.6 References[1] http://www.math.lsa.umich.edu/~mityab/teaching/m395f10/10_counterexamples.pdf

[2] Neeb, Karl-Hermann, On a theorem of S. Banach. J. Lie Theory 7 (1997), no. 2, 293–300.

This article incorporates material from Metrizable on PlanetMath, which is licensed under the Creative CommonsAttribution/Share-Alike License.

Chapter 32

Normal space

For normal vector space, see normal (geometry).

In topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T4:every two disjoint closed sets of X have disjoint open neighborhoods. A normal Hausdorff space is also called aT4 space. These conditions are examples of separation axioms and their further strengthenings define completelynormal Hausdorff spaces, or T5 spaces, and perfectly normal Hausdorff spaces, or T6 spaces.

32.1 Definitions

A topological space X is a normal space if, given any disjoint closed sets E and F, there are open neighbourhoodsU of E and V of F that are also disjoint. More intuitively, this condition says that E and F can be separated byneighbourhoods.

U

E

V

F

The closed sets E and F, here represented by closed disks on opposite sides of the picture, are separated by their respective neigh-bourhoods U and V, here represented by larger, but still disjoint, open disks.

A T4 space is a T1 space X that is normal; this is equivalent to X being normal and Hausdorff.A completely normal space or a hereditarily normal space is a topological space X such that every subspace of Xwith subspace topology is a normal space. It turns out that X is completely normal if and only if every two separatedsets can be separated by neighbourhoods.A completely T4 space, or T5 space is a completely normal T1 space topological space X, which implies that X is

135

136 CHAPTER 32. NORMAL SPACE

Hausdorff; equivalently, every subspace of X must be a T4 space.A perfectly normal space is a topological space X in which every two disjoint closed sets E and F can be preciselyseparated by a continuous function f fromX to the real lineR: the preimages of 0 and 1 under f are, respectively,E and F. (In this definition, the real line can be replaced with the unit interval [0,1].)It turns out that X is perfectly normal if and only if X is normal and every closed set is a Gδ set. Equivalently, X isperfectly normal if and only if every closed set is a zero set. Every perfectly normal space is automatically completelynormal.[1]

A Hausdorff perfectly normal space X is a T6 space, or perfectly T4 space.Note that the terms “normal space” and “T4" and derived concepts occasionally have a different meaning. (Nonethe-less, “T5" always means the same as “completely T4", whatever that may be.) The definitions given here are the onesusually used today. For more on this issue, see History of the separation axioms.Terms like “normal regular space" and “normal Hausdorff space” also turn up in the literature – they simply meanthat the space both is normal and satisfies the other condition mentioned. In particular, a normal Hausdorff space isthe same thing as a T4 space. Given the historical confusion of the meaning of the terms, verbal descriptions whenapplicable are helpful, that is, “normal Hausdorff” instead of “T4", or “completely normal Hausdorff” instead of “T5".Fully normal spaces and fully T4 spaces are discussed elsewhere; they are related to paracompactness.A locally normal space is a topological space where every point has an open neighbourhood that is normal. Everynormal space is locally normal, but the converse is not true. A classical example of a completely regular locallynormal space that is not normal is the Nemytskii plane.

32.2 Examples of normal spaces

Most spaces encountered in mathematical analysis are normal Hausdorff spaces, or at least normal regular spaces:

• All metric spaces (and hence all metrizable spaces) are perfectly normal Hausdorff;

• All pseudometric spaces (and hence all pseudometrisable spaces) are perfectly normal regular, although not ingeneral Hausdorff;

• All compact Hausdorff spaces are normal;

• In particular, the Stone–Čech compactification of a Tychonoff space is normal Hausdorff;

• Generalizing the above examples, all paracompact Hausdorff spaces are normal, and all paracompact regularspaces are normal;

• All paracompact topological manifolds are perfectly normal Hausdorff. However, there exist non-paracompactmanifolds which are not even normal.

• All order topologies on totally ordered sets are hereditarily normal and Hausdorff.

• Every regular second-countable space is completely normal, and every regular Lindelöf space is normal.

Also, all fully normal spaces are normal (even if not regular). Sierpinski space is an example of a normal space thatis not regular.

32.3 Examples of non-normal spaces

An important example of a non-normal topology is given by the Zariski topology on an algebraic variety or on thespectrum of a ring, which is used in algebraic geometry.A non-normal space of some relevance to analysis is the topological vector space of all functions from the real lineR to itself, with the topology of pointwise convergence. More generally, a theorem of A. H. Stone states that theproduct of uncountably many non-compact metric spaces is never normal.

32.4. PROPERTIES 137

32.4 Properties

Every closed subset of a normal space is normal. The continuous and closed image of a normal space is normal.[2]

The main significance of normal spaces lies in the fact that they admit “enough” continuous real-valued functions, asexpressed by the following theorems valid for any normal space X.Urysohn’s lemma: If A and B are two disjoint closed subsets of X, then there exists a continuous function f from Xto the real line R such that f(x) = 0 for all x in A and f(x) = 1 for all x in B. In fact, we can take the values of f to beentirely within the unit interval [0,1]. (In fancier terms, disjoint closed sets are not only separated by neighbourhoods,but also separated by a function.)More generally, the Tietze extension theorem: If A is a closed subset of X and f is a continuous function from A toR, then there exists a continuous function F: X→ R which extends f in the sense that F(x) = f(x) for all x in A.If U is a locally finite open cover of a normal space X, then there is a partition of unity precisely subordinate to U.(This shows the relationship of normal spaces to paracompactness.)In fact, any space that satisfies any one of these conditions must be normal.A product of normal spaces is not necessarily normal. This fact was first proved by Robert Sorgenfrey. An example ofthis phenomenon is the Sorgenfrey plane. Also, a subset of a normal space need not be normal (i.e. not every normalHausdorff space is a completely normal Hausdorff space), since every Tychonoff space is a subset of its Stone–Čechcompactification (which is normal Hausdorff). A more explicit example is the Tychonoff plank.

32.5 Relationships to other separation axioms

If a normal space is R0, then it is in fact completely regular. Thus, anything from “normal R0" to “normal completelyregular” is the same as what we normally call normal regular. Taking Kolmogorov quotients, we see that all normalT1 spaces are Tychonoff. These are what we normally call normal Hausdorff spaces.A topological space is said to be pseudonormal if given two disjoint closed sets in it, one of which is countable, thereare disjoint open sets containing them. Every normal space is pseudonormal, but not vice versa.Counterexamples to some variations on these statements can be found in the lists above. Specifically, Sierpinski spaceis normal but not regular, while the space of functions from R to itself is Tychonoff but not normal.

32.6 Citations[1] Munkres 2000, p. 213

[2] Willard, Stephen (1970). General topology. Reading, Mass.: Addison-Wesley Pub. Co. pp. 100–101. ISBN 0486434796.

32.7 References• Kemoto, Nobuyuki (2004). “Higher Separation Axioms”. In K.P. Hart, J. Nagata, and J.E. Vaughan. Ency-clopedia of General Topology. Amsterdam: Elsevier Science. ISBN 0-444-50355-2.

• Munkres, James R. (2000). Topology (2nd ed.). Prentice-Hall. ISBN 0-13-181629-2.

• Sorgenfrey, R.H. (1947). “On the topological product of paracompact spaces”. Bull. Amer. Math. Soc. 53:631–632. doi:10.1090/S0002-9904-1947-08858-3.

• Stone, A. H. (1948). “Paracompactness and product spaces”. Bull. Amer. Math. Soc. 54: 977–982.doi:10.1090/S0002-9904-1948-09118-2.

• Willard, Stephen (1970). General Topology. Reading, Massachusetts: Addison-Wesley. ISBN 0-486-43479-6.

Chapter 33

Open set

Example: The points (x, y) satisfying x2 + y2 = r2 are colored blue. The points (x, y) satisfying x2 + y2 < r2 are colored red. Thered points form an open set. The blue points form a boundary set. The union of the red and blue points is a closed set.

In topology, an open set is an abstract concept generalizing the idea of an open interval in the real line. The simplestexample is in metric spaces, where open sets can be defined as those sets which contain an open ball around each of

138

33.1. MOTIVATION 139

their points (or, equivalently, a set is open if it doesn't contain any of its boundary points); however, an open set, ingeneral, can be very abstract: any collection of sets can be called open, as long as the union of an arbitrary numberof open sets is open, the intersection of a finite number of open sets is open, and the space itself is open. Theseconditions are very loose, and they allow enormous flexibility in the choice of open sets. In the two extremes, everyset can be open (called the discrete topology), or no set can be open but the space itself (the indiscrete topology).In practice, however, open sets are usually chosen to be similar to the open intervals of the real line. The notion of anopen set provides a fundamental way to speak of nearness of points in a topological space, without explicitly havinga concept of distance defined. Once a choice of open sets is made, the properties of continuity, connectedness, andcompactness, which use notions of nearness, can be defined using these open sets.Each choice of open sets for a space is called a topology. Although open sets and the topologies that they comprise areof central importance in point-set topology, they are also used as an organizational tool in other important branchesof mathematics. Examples of topologies include the Zariski topology in algebraic geometry that reflects the algebraicnature of varieties, and the topology on a differential manifold in differential topology where each point within thespace is contained in an open set that is homeomorphic to an open ball in a finite-dimensional Euclidean space.

33.1 Motivation

Intuitively, an open set provides a method to distinguish two points. For example, if about one point in a topologicalspace there exists an open set not containing another (distinct) point, the two points are referred to as topologicallydistinguishable. In this manner, one may speak of whether two subsets of a topological space are “near” withoutconcretely defining a metric on the topological space. Therefore, topological spaces may be seen as a generalizationof metric spaces.In the set of all real numbers, one has the natural Euclidean metric; that is, a function which measures the distancebetween two real numbers: d(x, y) = |x - y|. Therefore, given a real number, one can speak of the set of all pointsclose to that real number; that is, within ε of that real number (refer to this real number as x). In essence, pointswithin ε of x approximate x to an accuracy of degree ε. Note that ε > 0 always but as ε becomes smaller and smaller,one obtains points that approximate x to a higher and higher degree of accuracy. For example, if x = 0 and ε = 1, thepoints within ε of x are precisely the points of the interval (−1, 1); that is, the set of all real numbers between −1and 1. However, with ε = 0.5, the points within ε of x are precisely the points of (−0.5, 0.5). Clearly, these pointsapproximate x to a greater degree of accuracy compared to when ε = 1.The previous discussion shows, for the case x = 0, that one may approximate x to higher and higher degrees ofaccuracy by defining ε to be smaller and smaller. In particular, sets of the form (-ε, ε) give us a lot of informationabout points close to x = 0. Thus, rather than speaking of a concrete Euclidean metric, one may use sets to describepoints close to x. This innovative idea has far-reaching consequences; in particular, by defining different collectionsof sets containing 0 (distinct from the sets (-ε, ε)), one may find different results regarding the distance between 0 andother real numbers. For example, if we were to define R as the only such set for “measuring distance”, all points areclose to 0 since there is only one possible degree of accuracy one may achieve in approximating 0: being a memberof R. Thus, we find that in some sense, every real number is distance 0 away from 0! It may help in this case to thinkof the measure as being a binary condition, all things in R are equally close to 0, while any item that is not in R is notclose to 0.In general, one refers to the family of sets containing 0, used to approximate 0, as a neighborhood basis; a memberof this neighborhood basis is referred to as an open set. In fact, one may generalize these notions to an arbitraryset (X); rather than just the real numbers. In this case, given a point (x) of that set, one may define a collectionof sets “around” (that is, containing) x, used to approximate x. Of course, this collection would have to satisfycertain properties (known as axioms) for otherwise we may not have a well-defined method to measure distance. Forexample, every point in X should approximate x to some degree of accuracy. Thus X should be in this family. Oncewe begin to define “smaller” sets containing x, we tend to approximate x to a greater degree of accuracy. Bearing thisin mind, one may define the remaining axioms that the family of sets about x is required to satisfy.

33.2 Definitions

The concept of open sets can be formalized with various degrees of generality, for example:

140 CHAPTER 33. OPEN SET

33.2.1 Euclidean space

A subset U of the Euclidean n-space Rn is called open if, given any point x in U, there exists a real number ε > 0 suchthat, given any point y in Rn whose Euclidean distance from x is smaller than ε, y also belongs to U.[1] Equivalently,a subset U of Rn is open if every point in U has a neighborhood in Rn contained in U.

33.2.2 Metric spaces

A subset U of a metric space (M, d) is called open if, given any point x in U, there exists a real number ε > 0 suchthat, given any point y in M with d(x, y) < ε, y also belongs to U. Equivalently, U is open if every point in U has aneighbourhood contained in U.This generalises the Euclidean space example, since Euclidean space with the Euclidean distance is a metric space.

33.2.3 Topological spaces

In general topological spaces, the open sets can be almost anything, with different choices giving different spaces.Let X be a set and τ be a family of sets. We say that τ is a topology on X if:

• X ∈ τ, ∅ ∈ τ ( X and ∅ are in τ )

• Oii∈I ⊆ τ ⇒ ∪i∈IOi ∈ τ (any union of sets in τ is in τ )

• Oii∈I ⊆ τ ⇒ ∩ni=1Oi ∈ τ (any finite intersection of sets in τ is in τ )

We call the sets in τ the open sets.Note that infinite intersections of open sets need not be open. For example, the intersection of all intervals of theform (−1/n, 1/n), where n is a positive integer, is the set 0 which is not open in the real line. Sets that can beconstructed as the intersection of countably many open sets are denoted Gδ sets.The topological definition of open sets generalises the metric space definition: If one begins with a metric space anddefines open sets as before, then the family of all open sets is a topology on the metric space. Every metric space istherefore, in a natural way, a topological space. There are, however, topological spaces that are not metric spaces.

33.3 Properties

• The empty set is both open and closed (clopen set).[2]

• The set X that the topology is defined on is both open and closed (clopen set).

• The union of any number of open sets is open.[3]

• The intersection of a finite number of open sets is open.[3]

33.4 Uses

Open sets have a fundamental importance in topology. The concept is required to define andmake sense of topologicalspace and other topological structures that deal with the notions of closeness and convergence for spaces such as metricspaces and uniform spaces.Every subset A of a topological space X contains a (possibly empty) open set; the largest such open set is called theinterior of A. It can be constructed by taking the union of all the open sets contained in A.Given topological spaces X and Y, a function f from X to Y is continuous if the preimage of every open set in Y isopen in X. The function f is called open if the image of every open set in X is open in Y.An open set on the real line has the characteristic property that it is a countable union of disjoint open intervals.

33.5. NOTES AND CAUTIONS 141

33.5 Notes and cautions

33.5.1 “Open” is defined relative to a particular topology

Whether a set is open depends on the topology under consideration. Having opted for greater brevity over greaterclarity, we refer to a set X endowed with a topology T as “the topological space X" rather than “the topological space(X, T)", despite the fact that all the topological data is contained in T. If there are two topologies on the same set, a setU that is open in the first topology might fail to be open in the second topology. For example, if X is any topologicalspace and Y is any subset of X, the set Y can be given its own topology (called the 'subspace topology') defined by“a set U is open in the subspace topology on Y if and only if U is the intersection of Y with an open set from theoriginal topology on X.” This potentially introduces new open sets: if V is open in the original topology on X, butV ∩ Y isn't, then V ∩ Y is open in the subspace topology on Y but not in the original topology on X.As a concrete example of this, if U is defined as the set of rational numbers in the interval (0, 1), then U is an opensubset of the rational numbers, but not of the real numbers. This is because when the surrounding space is the rationalnumbers, for every point x in U, there exists a positive number a such that all rational points within distance a ofx are also in U. On the other hand, when the surrounding space is the reals, then for every point x in U there is nopositive a such that all real points within distance a of x are in U (since U contains no non-rational numbers).

33.5.2 Open and closed are not mutually exclusive

A set might be open, closed, both, or neither.For example, we'll use the real line with its usual topology (the Euclidean topology), which is defined as follows:every interval (a,b) of real numbers belongs to the topology, and every union of such intervals, e.g. (a, b) ∪ (c, d) ,belongs to the topology.

• In any topology, the entire set X is declared open by definition, as is the empty set. Moreover, the complementof the entire set X is the empty set; since X has an open complement, this means by definition that X is closed.Hence, in any topology, the entire space is simultaneously open and closed ("clopen").

• The interval I = (0, 1) is open because it belongs to the Euclidean topology. If I were to have an opencomplement, it would mean by definition that I were closed. But I does not have an open complement; itscomplement is IC = (−∞, 0] ∪ [1,∞) , which does not belong to the Euclidean topology since it is not aunion of intervals of the form (a, b) . Hence, I is an example of a set that is open but not closed.

• By a similar argument, the interval J = [0, 1] is closed but not open.• Finally, since neither K = [0, 1) nor its complement KC = (−∞, 0) ∪ [1,∞) belongs to the Euclideantopology (neither one can be written as a union of intervals of the form (a,b) ), this means that K is neitheropen nor closed.

33.6 See also• Closed set• Clopen set• Neighbourhood

33.7 References[1] Ueno, Kenji et al. (2005). “The birth of manifolds”. A Mathematical Gift: The Interplay Between Topology, Functions,

Geometry, and Algebra. Vol. 3. American Mathematical Society. p. 38. ISBN 9780821832844.

[2] Krantz, Steven G. (2009). “Fundamentals”. Essentials of Topology With Applications. CRC Press. pp. 3–4. ISBN9781420089745.

[3] Taylor, Joseph L. (2011). “Analytic functions”. Complex Variables. The Sally Series. American Mathematical Society. p.29. ISBN 9780821869017.

142 CHAPTER 33. OPEN SET

33.8 External links• Hazewinkel, Michiel, ed. (2001), “Open set”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Open Set at PlanetMath.org.

Chapter 34

Order theory

For a topical guide to this subject, see Outline of order theory.

Order theory is a branch of mathematics which investigates our intuitive notion of order using binary relations. Itprovides a formal framework for describing statements such as “this is less than that” or “this precedes that”. Thisarticle introduces the field and provides basic definitions. A list of order-theoretic terms can be found in the ordertheory glossary.

34.1 Background and motivation

Orders are everywhere in mathematics and related fields like computer science. The first order often discussed inprimary school is the standard order on the natural numbers e.g. “2 is less than 3”, “10 is greater than 5”, or “DoesTom have fewer cookies than Sally?". This intuitive concept can be extended to orders on other sets of numbers,such as the integers and the reals. The idea of being greater than or less than another number is one of the basicintuitions of number systems (compare with numeral systems) in general (although one usually is also interested inthe actual difference of two numbers, which is not given by the order). Another familiar example of an ordering isthe lexicographic order of words in a dictionary.The above types of orders have a special property: each element can be compared to any other element, i.e. it isgreater, smaller, or equal. However, this is not always a desired requirement. For example, consider the subsetordering of sets. If a set A contains all the elements of a set B, then B is said to be smaller than or equal to A. Yetthere are some sets that cannot be related in this fashion. Whenever both contain some elements that are not in theother, the two sets are not related by subset-inclusion. Hence, subset-inclusion is only a partial order, as opposed tothe total orders given before.Order theory captures the intuition of orders that arises from such examples in a general setting. This is achievedby specifying properties that a relation ≤ must have to be a mathematical order. This more abstract approach makesmuch sense, because one can derive numerous theorems in the general setting, without focusing on the details of anyparticular order. These insights can then be readily transferred to many less abstract applications.Driven by the wide practical usage of orders, numerous special kinds of ordered sets have been defined, some ofwhich have grown into mathematical fields of their own. In addition, order theory does not restrict itself to thevarious classes of ordering relations, but also considers appropriate functions between them. A simple example of anorder theoretic property for functions comes from analysis where monotone functions are frequently found.

34.2 Basic definitions

This section introduces ordered sets by building upon the concepts of set theory, arithmetic, and binary relations.

143

144 CHAPTER 34. ORDER THEORY

34.2.1 Partially ordered sets

Orders are special binary relations. Suppose that P is a set and that ≤ is a relation on P. Then ≤ is a partial order ifit is reflexive, antisymmetric, and transitive, i.e., for all a, b and c in P, we have that:

a ≤ a (reflexivity)if a ≤ b and b ≤ a then a = b (antisymmetry)if a ≤ b and b ≤ c then a ≤ c (transitivity).

A set with a partial order on it is called a partially ordered set, poset, or just an ordered set if the intended meaningis clear. By checking these properties, one immediately sees that the well-known orders on natural numbers, integers,rational numbers and reals are all orders in the above sense. However, they have the additional property of beingtotal, i.e., for all a and b in P, we have that:

a ≤ b or b ≤ a (totality).

These orders can also be called linear orders or chains. While many classical orders are linear, the subset order onsets provides an example where this is not the case. Another example is given by the divisibility relation "|". For twonatural numbers n and m, we write n|m if n divides m without remainder. One easily sees that this yields a partialorder. The identity relation = on any set is also a partial order in which every two distinct elements are incomparable.It is also the only relation that is both a partial order and an equivalence relation. Many advanced properties of posetsare interesting mainly for non-linear orders.

34.2.2 Visualizing a poset

Hasse diagrams can visually represent the elements and relations of a partial ordering. These are graph drawingswhere the vertices are the elements of the poset and the ordering relation is indicated by both the edges and therelative positioning of the vertices. Orders are drawn bottom-up: if an element x is smaller than (precedes) y thenthere exists a path from x to y that is directed upwards. It is often necessary for the edges connecting elements tocross each other, but elements must never be located within an edge. An instructive exercise is to draw the Hassediagram for the set of natural numbers that are smaller than or equal to 13, ordered by | (the divides relation).Even some infinite sets can be diagrammed by superimposing an ellipsis (...) on a finite sub-order. This works wellfor the natural numbers, but it fails for the reals, where there is no immediate successor above 0; however, quite oftenone can obtain an intuition related to diagrams of a similar kind.

34.2.3 Special elements within an order

In a partially ordered set there may be some elements that play a special role. The most basic example is given by theleast element of a poset. For example, 1 is the least element of the positive integers and the empty set is the least setunder the subset order. Formally, an element m is a least element if:

m ≤ a, for all elements a of the order.

The notation 0 is frequently found for the least element, even when no numbers are concerned. However, in orderson sets of numbers, this notation might be inappropriate or ambiguous, since the number 0 is not always least. Anexample is given by the above divisibility order |, where 1 is the least element since it divides all other numbers. Incontrast, 0 is the number that is divided by all other numbers. Hence it is the greatest element of the order. Otherfrequent terms for the least and greatest elements is bottom and top or zero and unit.Least and greatest elements may fail to exist, as the example of the real numbers shows. But if they exist, they arealways unique. In contrast, consider the divisibility relation | on the set 2,3,4,5,6. Although this set has neither topnor bottom, the elements 2, 3, and 5 have no elements below them, while 4, 5, and 6 have none above. Such elementsare calledminimal andmaximal, respectively. Formally, an element m is minimal if:

a ≤ m implies a = m, for all elements a of the order.

34.2. BASIC DEFINITIONS 145

1

2

461015

30 20

60

12

53

Hasse diagram of the set of all divisors of 60, partially ordered by divisibility

Exchanging ≤ with ≥ yields the definition of maximality. As the example shows, there can be manymaximal elementsand some elements may be both maximal and minimal (e.g. 5 above). However, if there is a least element, then itis the only minimal element of the order. Again, in infinite posets maximal elements do not always exist - the setof all finite subsets of a given infinite set, ordered by subset inclusion, provides one of many counterexamples. Animportant tool to ensure the existence of maximal elements under certain conditions is Zorn’s Lemma.Subsets of partially ordered sets inherit the order. We already applied this by considering the subset 2,3,4,5,6 ofthe natural numbers with the induced divisibility ordering. Now there are also elements of a poset that are specialwith respect to some subset of the order. This leads to the definition of upper bounds. Given a subset S of someposet P, an upper bound of S is an element b of P that is above all elements of S. Formally, this means that

s ≤ b, for all s in S.

Lower bounds again are defined by inverting the order. For example, −5 is a lower bound of the natural numbersas a subset of the integers. Given a set of sets, an upper bound for these sets under the subset ordering is given bytheir union. In fact, this upper bound is quite special: it is the smallest set that contains all of the sets. Hence, wehave found the least upper bound of a set of sets. This concept is also called supremum or join, and for a set Sone writes sup(S) or vS for its least upper bound. Conversely, the greatest lower bound is known as infimum ormeet and denoted inf(S) or ^S. These concepts play an important role in many applications of order theory. For twoelements x and y, one also writes x v y and x ^ y for sup(x,y) and inf(x,y), respectively.For another example, consider again the relation | on natural numbers. The least upper bound of two numbers is thesmallest number that is divided by both of them, i.e. the least common multiple of the numbers. Greatest lowerbounds in turn are given by the greatest common divisor.

146 CHAPTER 34. ORDER THEORY

34.2.4 Duality

In the previous definitions, we often noted that a concept can be defined by just inverting the ordering in a formerdefinition. This is the case for “least” and “greatest”, for “minimal” and “maximal”, for “upper bound” and “lowerbound”, and so on. This is a general situation in order theory: A given order can be inverted by just exchangingits direction, pictorially flipping the Hasse diagram top-down. This yields the so-called dual, inverse, or oppositeorder.Every order theoretic definition has its dual: it is the notion one obtains by applying the definition to the inverse order.Since all concepts are symmetric, this operation preserves the theorems of partial orders. For a given mathematicalresult, one can just invert the order and replace all definitions by their duals and one obtains another valid theorem.This is important and useful, since one obtains two theorems for the price of one. Some more details and examplescan be found in the article on duality in order theory.

34.2.5 Constructing new orders

There are many ways to construct orders out of given orders. The dual order is one example. Another importantconstruction is the cartesian product of two partially ordered sets, taken together with the product order on pairs ofelements. The ordering is defined by (a, x) ≤ (b, y) if (and only if) a ≤ b and x ≤ y. (Notice carefully that thereare three distinct meanings for the relation symbol ≤ in this definition.) The disjoint union of two posets is anothertypical example of order construction, where the order is just the (disjoint) union of the original orders.Every partial order ≤ gives rise to a so-called strict order <, by defining a < b if a≤ b and not b≤ a. This transformationcan be inverted by setting a ≤ b if a < b or a = b. The two concepts are equivalent although in some circumstancesone can be more convenient to work with than the other.

34.3 Functions between orders

It is reasonable to consider functions between partially ordered sets having certain additional properties that are relatedto the ordering relations of the two sets. The most fundamental condition that occurs in this context is monotonicity.A function f from a poset P to a poset Q is monotone, or order-preserving, if a ≤ b in P implies f(a) ≤ f(b) inQ (Noting that, strictly, the two relations here are different since they apply to different sets.). The converse of thisimplication leads to functions that are order-reflecting, i.e. functions f as above for which f(a) ≤ f(b) implies a ≤b. On the other hand, a function may also be order-reversing or antitone, if a ≤ b implies f(b) ≤ f(a).An order-embedding is a function f between orders that is both order-preserving and order-reflecting. Examplesfor these definitions are found easily. For instance, the function that maps a natural number to its successor is clearlymonotone with respect to the natural order. Any function from a discrete order, i.e. from a set ordered by the identityorder "=", is also monotone. Mapping each natural number to the corresponding real number gives an example foran order embedding. The set complement on a powerset is an example of an antitone function.An important question is when two orders are “essentially equal”, i.e. when they are the same up to renaming ofelements. Order isomorphisms are functions that define such a renaming. An order-isomorphism is a monotonebijective function that has a monotone inverse. This is equivalent to being a surjective order-embedding. Hence, theimage f(P) of an order-embedding is always isomorphic to P, which justifies the term “embedding”.A more elaborate type of functions is given by so-called Galois connections. Monotone Galois connections canbe viewed as a generalization of order-isomorphisms, since they constitute of a pair of two functions in conversedirections, which are “not quite” inverse to each other, but that still have close relationships.Another special type of self-maps on a poset are closure operators, which are not onlymonotonic, but also idempotent,i.e. f(x) = f(f(x)), and extensive (or inflationary), i.e. x ≤ f(x). These have many applications in all kinds of “clo-sures” that appear in mathematics.Besides being compatible with the mere order relations, functions between posets may also behave well with respectto special elements and constructions. For example, when talking about posets with least element, it may seemreasonable to consider only monotonic functions that preserve this element, i.e. which map least elements to leastelements. If binary infima ^ exist, then a reasonable property might be to require that f(x^y) = f(x)^f(y), for all x andy. All of these properties, and indeed many more, may be compiled under the label of limit-preserving functions.Finally, one can invert the view, switching from functions of orders to orders of functions. Indeed, the functions

34.4. SPECIAL TYPES OF ORDERS 147

between two posets P and Q can be ordered via the pointwise order. For two functions f and g, we have f ≤ g if f(x)≤ g(x) for all elements x of P. This occurs for example in domain theory, where function spaces play an importantrole.

34.4 Special types of orders

Many of the structures that are studied in order theory employ order relations with further properties. In fact, evensome relations that are not partial orders are of special interest. Mainly the concept of a preorder has to be mentioned.A preorder is a relation that is reflexive and transitive, but not necessarily antisymmetric. Each preorder induces anequivalence relation between elements, where a is equivalent to b, if a ≤ b and b ≤ a. Preorders can be turned intoorders by identifying all elements that are equivalent with respect to this relation.Several types of orders can be defined from numerical data on the items of the order: a total order results fromattaching distinct real numbers to each item and using the numerical comparisons to order the items; instead, ifdistinct items are allowed to have equal numerical scores, one obtains a strict weak ordering. Requiring two scores tobe separated by a fixed threshold before they may be compared leads to the concept of a semiorder, while allowingthe threshold to vary on a per-item basis produces an interval order.An additional simple but useful property leads to so-called well-orders, for which all non-empty subsets have aminimal element. Generalizing well-orders from linear to partial orders, a set is well partially ordered if all itsnon-empty subsets have a finite number of minimal elements.Many other types of orders arise when the existence of infima and suprema of certain sets is guaranteed. Focusingon this aspect, usually referred to as completeness of orders, one obtains:

• Bounded posets, i.e. posets with a least and greatest element (which are just the supremum and infimum ofthe empty subset),

• Lattices, in which every non-empty finite set has a supremum and infimum,

• Complete lattices, where every set has a supremum and infimum, and

• Directed complete partial orders (dcpos), that guarantee the existence of suprema of all directed subsets andthat are studied in domain theory.

• Partial orders with complements, or poc sets,[1] are posets S having a unique bottom element 0∈S, along withan order-reversing involution, such that a ≤ a∗ ⇒ a = 0 .

However, one can go even further: if all finite non-empty infima exist, then ∧ can be viewed as a total binary operationin the sense of universal algebra. Hence, in a lattice, two operations ∧ and ∨ are available, and one can define newproperties by giving identities, such as

x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z), for all x, y, and z.

This condition is called distributivity and gives rise to distributive lattices. There are some other important distribu-tivity laws which are discussed in the article on distributivity in order theory. Some additional order structures thatare often specified via algebraic operations and defining identities are

• Heyting algebras and

• Boolean algebras,

which both introduce a new operation ~ called negation. Both structures play a role in mathematical logic andespecially Boolean algebras have major applications in computer science. Finally, various structures in mathematicscombine orders with even more algebraic operations, as in the case of quantales, that allow for the definition of anaddition operation.Many other important properties of posets exist. For example, a poset is locally finite if every closed interval [a,b] in it is finite. Locally finite posets give rise to incidence algebras which in turn can be used to define the Eulercharacteristic of finite bounded posets.

148 CHAPTER 34. ORDER THEORY

34.5 Subsets of ordered sets

In an ordered set, one can define many types of special subsets based on the given order. A simple example are uppersets; i.e. sets that contain all elements that are above them in the order. Formally, the upper closure of a set S in aposet P is given by the set x in P | there is some y in S with y ≤ x. A set that is equal to its upper closure is calledan upper set. Lower sets are defined dually.More complicated lower subsets are ideals, which have the additional property that each two of their elements havean upper bound within the ideal. Their duals are given by filters. A related concept is that of a directed subset, whichlike an ideal contains upper bounds of finite subsets, but does not have to be a lower set. Furthermore it is oftengeneralized to preordered sets.A subset which is - as a sub-poset - linearly ordered, is called a chain. The opposite notion, the antichain, is a subsetthat contains no two comparable elements; i.e. that is a discrete order.

34.6 Related mathematical areas

Although most mathematical areas use orders in one or the other way, there are also a few theories that have relation-ships which go far beyond mere application. Together with their major points of contact with order theory, some ofthese are to be presented below.

34.6.1 Universal algebra

As alreadymentioned, the methods and formalisms of universal algebra are an important tool for many order theoreticconsiderations. Beside formalizing orders in terms of algebraic structures that satisfy certain identities, one can alsoestablish other connections to algebra. An example is given by the correspondence between Boolean algebras andBoolean rings. Other issues are concerned with the existence of free constructions, such as free lattices based on agiven set of generators. Furthermore, closure operators are important in the study of universal algebra.

34.6.2 Topology

In topology orders play a very prominent role. In fact, the set of open sets provides a classical example of a completelattice, more precisely a complete Heyting algebra (or "frame" or "locale"). Filters and nets are notions closely relatedto order theory and the closure operator of sets can be used to define topology. Beyond these relations, topology canbe looked at solely in terms of the open set lattices, which leads to the study of pointless topology. Furthermore, anatural preorder of elements of the underlying set of a topology is given by the so-called specialization order, that isactually a partial order if the topology is T0.Conversely, in order theory, one often makes use of topological results. There are various ways to define subsets ofan order which can be considered as open sets of a topology. Especially, it is interesting to consider topologies on aposet (X, ≤) that in turn induce ≤ as their specialization order. The finest such topology is the Alexandrov topology,given by taking all upper sets as opens. Conversely, the coarsest topology that induces the specialization order is theupper topology, having the complements of principal ideals (i.e. sets of the form y in X | y ≤ x for some x) as asubbase. Additionally, a topology with specialization order ≤ may be order consistent, meaning that their open setsare “inaccessible by directed suprema” (with respect to ≤). The finest order consistent topology is the Scott topology,which is coarser than the Alexandrov topology. A third important topology in this spirit is the Lawson topology.There are close connections between these topologies and the concepts of order theory. For example, a functionpreserves directed suprema iff it is continuous with respect to the Scott topology (for this reason this order theoreticproperty is also called Scott-continuity).

34.6.3 Category theory

The visualization of orders with Hasse diagrams has a straightforward generalization: instead of displaying lesserelements below greater ones, the direction of the order can also be depicted by giving directions to the edges of agraph. In this way, each order is seen to be equivalent to a directed acyclic graph, where the nodes are the elements

34.7. HISTORY 149

of the poset and there is a directed path from a to b if and only if a ≤ b. Dropping the requirement of being acyclic,one can also obtain all preorders.When equipped with all transitive edges, these graphs in turn are just special categories, where elements are objectsand each set of morphisms between two elements is at most singleton. Functions between orders become functorsbetween categories. Interestingly, many ideas of order theory are just concepts of category theory in small. Forexample, an infimum is just a categorical product. More generally, one can capture infima and suprema under theabstract notion of a categorical limit (or colimit, respectively). Another place where categorical ideas occur is theconcept of a (monotone) Galois connection, which is just the same as a pair of adjoint functors.But category theory also has its impact on order theory on a larger scale. Classes of posets with appropriate functionsas discussed above form interesting categories. Often one can also state constructions of orders, like the productorder, in terms of categories. Further insights result when categories of orders are found categorically equivalent toother categories, for example of topological spaces. This line of research leads to various representation theorems,often collected under the label of Stone duality.

34.7 History

As explained before, orders are ubiquitous inmathematics. However, earliest explicit mentionings of partial orders areprobably to be found not before the 19th century. In this context the works of George Boole are of great importance.Moreover, works of Charles Sanders Peirce, Richard Dedekind, and Ernst Schröder also consider concepts of ordertheory. Certainly, there are others to be named in this context and surely there exists more detailed material on thehistory of order theory.The term poset as an abbreviation for partially ordered set was coined by Garrett Birkhoff in the second edition ofhis influential book Lattice Theory.[2][3]

34.8 See also• Cyclic order

• Hierarchy

• Incidence algebra

• Important publications in order theory

• Causal Sets

34.9 Notes[1] Roller, Martin A. (1998), Poc sets, median algebras and group actions. An extended study of Dunwoody’s construction and

Sageev’s theorem (PDF), Southampton Preprint Archive

[2] Birkhoff 1948, p.1

[3] Earliest Known Uses of Some of the Words of Mathematics

34.10 References• Birkhoff, Garrett (1940). Lattice Theory 25 (3rd Revised ed.). American Mathematical Society. ISBN 978-0-8218-1025-5.

• Burris, S. N.; Sankappanavar, H. P. (1981). ACourse in Universal Algebra. Springer. ISBN 978-0-387-90578-5.

• Davey, B. A.; Priestley, H. A. (2002). Introduction to Lattices and Order (2nd ed.). Cambridge UniversityPress. ISBN 0-521-78451-4.

150 CHAPTER 34. ORDER THEORY

• Gierz, G.; Hofmann, K. H.; Keimel, K.; Mislove, M.; Scott, D. S. (2003). Continuous Lattices and Domains.Encyclopedia of Mathematics and its Applications 93. Cambridge University Press. ISBN 978-0-521-80338-0.

34.11 External links• Orders at ProvenMath partial order, linear order, well order, initial segment; formal definitions and proofswithin the axioms of set theory.

• Nagel, Felix (2013). Set Theory and Topology. An Introduction to the Foundations of Analysis

Chapter 35

Orthocompact space

In mathematics, in the field of general topology, a topological space is said to be orthocompact if every open coverhas an interior preserving open refinement. That is, given an open cover of the topological space, there is a refinementwhich is also an open cover, with the further property that at any point, the intersection of all open sets in the refinementcontaining that point, is also open.If the number of open sets containing the point is finite, then their intersection is clearly open. That is, every pointfinite open cover is interior preserving. Hence, we have the following: every metacompact space, and in particular,every paracompact space, is orthocompact.Useful theorems:

• Orthocompactness is a topological invariant; that is, it is preserved by homeomorphisms.

• Every closed subspace of an orthocompact space is orthocompact.

• A topological space X is orthocompact if and only if every open cover of X by basic open subsets of X has aninterior-preserving refinement that is an open cover of X.

• The product X × [0,1] of the closed unit interval with an orthocompact space X is orthocompact if and only ifX is countably metacompact. (B.M. Scott) [1]

• Every orthocompact space is countably orthocompact.

• Every countably orthocompact Lindelöf space is orthocompact.

35.1 References[1] B.M. Scott, Towards a product theory for orthocompactness, “Studies in Topology”, N.M. Stavrakas and K.R. Allen, eds

(1975), 517–537.

• P. Fletcher, W.F. Lindgren, Quasi-uniform Spaces, Marcel Dekker, 1982, ISBN 0-8247-1839-9. Chap.V.

151

Chapter 36

Paracompact space

In mathematics, a paracompact space is a topological space in which every open cover has an open refinement thatis locally finite. These spaces were introduced by Dieudonné (1944). Every compact space is paracompact. Everyparacompact Hausdorff space is normal, and a Hausdorff space is paracompact if and only if it admits partitions ofunity subordinate to any open cover. Sometimes paracompact spaces are defined so as to always be Hausdorff.Every closed subspace of a paracompact space is paracompact. While compact subsets of Hausdorff spaces arealways closed, this is not true for paracompact subsets. A space such that every subspace of it is a paracompact spaceis called hereditarily paracompact. This is equivalent to requiring that every open subspace be paracompact.Tychonoff’s theorem (which states that the product of any collection of compact topological spaces is compact) doesnot generalize to paracompact spaces in that the product of paracompact spaces need not be paracompact. However,the product of a paracompact space and a compact space is always paracompact.Every metric space is paracompact. A topological space is metrizable if and only if it is a paracompact and locallymetrizable Hausdorff space.

36.1 Paracompactness

A cover of a set X is a collection of subsets of X whose union contains X. In symbols, if U = Uα : α in A is anindexed family of subsets of X, then U is a cover of X if

X ⊆∪α∈A

Uα.

A cover of a topological space X is open if all its members are open sets. A refinement of a cover of a space X is anew cover of the same space such that every set in the new cover is a subset of some set in the old cover. In symbols,the cover V = Vᵦ : β in B is a refinement of the cover U = Uα : α in A if and only if, for any Vᵦ in V, thereexists some Uα in U such that Vᵦ⊆Uα.An open cover of a space X is locally finite if every point of the space has a neighborhood that intersects only finitelymany sets in the cover. In symbols, U = Uα : α in A is locally finite if and only if, for any x in X, there exists someneighbourhood V(x) of x such that the set

α ∈ A : Uα ∩ V (x) = ∅

is finite. A topological spaceX is now said to be paracompact if every open cover has a locally finite open refinement.

36.2 Examples• Every compact space is paracompact.

152

36.3. PROPERTIES 153

• Every regular Lindelöf space is paracompact. In particular, every locally compact Hausdorff second-countablespace is paracompact.

• The Sorgenfrey line is paracompact, even though it is neither compact, locally compact, second countable, normetrizable.

• Every CW complex is paracompact [1]

• (Theorem of A. H. Stone) Every metric space is paracompact.[2] Early proofs were somewhat involved, butan elementary one was found by M. E. Rudin.[3] Existing proofs of this require the axiom of choice for thenon-separable case. It has been shown that neither ZF theory nor ZF theory with the axiom of dependentchoice is sufficient.[4]

Some examples of spaces that are not paracompact include:

• The most famous counterexample is the long line, which is a nonparacompact topological manifold. (The longline is locally compact, but not second countable.)

• Another counterexample is a product of uncountably many copies of an infinite discrete space. Any infiniteset carrying the particular point topology is not paracompact; in fact it is not even metacompact.

• The Prüfer manifold is a non-paracompact surface.

• The bagpipe theorem shows that there are 2ℵ1 isomorphism classes of non-paracompact surfaces.

36.3 Properties

Paracompactness is weakly hereditary, i.e. every closed subspace of a paracompact space is paracompact. This canbe extended to F-sigma subspaces as well.

• A regular space is paracompact if every open cover admits a locally finite refinement. (Here, the refinement isnot required to be open.) In particular, every regular Lindelof space is paracompact.

• (Smirnov metrization theorem) A topological space is metrizable if and only if it is paracompact, Hausdorff,and locally metrizable.

• Michael selection theorem states that lower semicontinuousmultifunctions fromX into nonempty closed convexsubsets of Banach spaces admit continuous selection iff X is paracompact.

Although a product of paracompact spaces need not be paracompact, the following are true:

• The product of a paracompact space and a compact space is paracompact.

• The product of a metacompact space and a compact space is metacompact.

Both these results can be proved by the tube lemma which is used in the proof that a product of finitely many compactspaces is compact.

36.4 Paracompact Hausdorff Spaces

Paracompact spaces are sometimes required to also be Hausdorff to extend their properties.

• (Theorem of Jean Dieudonné) Every paracompact Hausdorff space is normal.

• Every paracompact Hausdorff space is a shrinking space, that is, every open cover of a paracompact Hausdorffspace has a shrinking: another open cover indexed by the same set such that the closure of every set in the newcover lies inside the corresponding set in the old cover.

• On paracompact Hausdorff spaces, sheaf cohomology and Čech cohomology are equal.[5]

154 CHAPTER 36. PARACOMPACT SPACE

36.4.1 Partitions of unity

The most important feature of paracompact Hausdorff spaces is that they are normal and admit partitions of unitysubordinate to any open cover. This means the following: if X is a paracompact Hausdorff space with a given opencover, then there exists a collection of continuous functions on X with values in the unit interval [0, 1] such that:

• for every function f: X → R from the collection, there is an open set U from the cover such that the supportof f is contained in U;

• for every point x in X, there is a neighborhood V of x such that all but finitely many of the functions in thecollection are identically 0 in V and the sum of the nonzero functions is identically 1 in V.

In fact, a T1 space is Hausdorff and paracompact if and only if it admits partitions of unity subordinate to any opencover (see below). This property is sometimes used to define paracompact spaces (at least in the Hausdorff case).Partitions of unity are useful because they often allow one to extend local constructions to the whole space. Forinstance, the integral of differential forms on paracompact manifolds is first defined locally (where the manifold lookslike Euclidean space and the integral is well known), and this definition is then extended to the whole space via apartition of unity.

Proof that paracompact Hausdorff spaces admit partitions of unity

A Hausdorff space X is paracompact if and only if it every open cover admits a subordinate partition of unity. Theif direction is straightforward. Now for the only if direction, we do this in a few stages.

Lemma 1: If O is a locally finite open cover, then there exists open setsWU for each U ∈ O , suchthat each WU ⊆ U and WU : U ∈ O is a locally finite refinement.

Lemma 2: IfO is a locally finite open cover, then there are continuous functions fU : X → [0, 1] suchthat supp fU ⊆ U and such that f :=

∑U∈O fU is a continuous function which is always non-zero

and finite.

Theorem: In a paracompact Hausdorff spaceX , ifO is an open cover, then there exists a partition ofunity subordinate to it.

Proof (Lemma 1): Let V be the collection of open sets meeting only finitely many sets in O , andwhose closure is contained in a set in O . One can check as an exercise that this provides an openrefinement, since paracompact Hausdorff spaces are regular, and since O is locally finite. Now replaceV by a locally finite open refinement. One can easily check that each set in this refinement has the sameproperty as that which characterised the original cover.

Now we defineWU =∪A ∈ V : A ⊆ U . We have that each WU ⊆ U ; for otherwise: suppose

there is x ∈ WU \ U . We will show that there is closed set C ⊃ WU such that x /∈ C (this meanssimply x /∈ WU by definition of closure). Since we chose V to be locally finite there is neighbourhoodV [x] of x such that only finitely many sets U1, ..., Un ∈ A ∈ V : A ⊆ U have non-empty intersectionwith V [x] . We take their closures U1, ..., Un and then V := V [x] \ ∪Ui is an open set (since sum isfinite) such that V ∩WU = ∅ . Moreover x ∈ V , because ∀i = 1, ..., n we have Ui ⊆ U and weknow that x /∈ U . Then C := X \ V is closed set without x which conatins WU . So x /∈ WU andwe've reached contradiction. And it easy to see that WU : U ∈ O is an open refinement of O .

Finally, to verify that this cover is locally finite, fix x ∈ X and letN be a neighbourhood of x . We knowthat for each U we haveWU ⊆ U . Since O is locally finite there are only finitely many sets U1, ..., Uk

having non-empty intersection withN . Then only setsWU1 , ...,WUkhave non-empty intersection with

N , because for every other U ′ we have N ∩WU ′ ⊆ N ∩ U ′ = ∅

36.5. RELATIONSHIP WITH COMPACTNESS 155

Proof (Lemma 2): Applying Lemma 1, let fU : X → [0, 1] be coninuous maps with fU WU = 1and supp fU ⊆ U (by Urysohn’s lemma for disjoint closed sets in normal spaces, which a paracompactHausdorff space is). Note by the support of a function, we here mean the points not mapping to zero(and not the closure of this set). To show that f =

∑U∈O fU is always finite and non-zero, take x ∈ X

, and let N a neighbourhood of x meeting only finitely many sets in O ; thus x belongs to only finitelymany sets in O ; thus fU (x) = 0 for all but finitely many U ; moreover x ∈ WU for some U ,thus fU (x) = 1 ; so f(x) is finite and ≥ 1 . To establish continuity, take x,N as before, and letS = U ∈ O : N meets U , which is finite; then f N =

∑U∈S fU N , which is a continuous

function; hence the preimage under f of a neighbourhood of f(x) will be a neighbourhood of x .

Proof (Theorem): TakeO∗ a locally finite subcover of the refinement cover: V open : (∃U ∈ O)V ⊆U . Applying Lemma 2, we obtain continuous functions fW : X → [0, 1] with supp fW ⊆W (thusthe usual closed version of the support is contained in some U ∈ O , for eachW ∈ O∗ ; for which theirsum constitutes a continuous function which is always finite non-zero (hence 1/f is continuous positive,finite-valued). So replacing each fW by fW /f , we have now — all things remaining the same — thattheir sum is everywhere 1 . Finally for x ∈ X , lettingN be a neighbourhood of x meeting only finitelymany sets in O∗ , we have fW N = 0 for all but finitely manyW ∈ O∗ since each supp fW ⊆W. Thus we have a partition of unity subordinate to the original open cover.

36.5 Relationship with compactness

There is a similarity between the definitions of compactness and paracompactness: For paracompactness, “subcover”is replaced by “open refinement” and “finite” by is replaced by “locally finite”. Both of these changes are significant:if we take the definition of paracompact and change “open refinement” back to “subcover”, or “locally finite” back to“finite”, we end up with the compact spaces in both cases.Paracompactness has little to do with the notion of compactness, but rather more to do with breaking up topologicalspace entities into manageable pieces.

36.5.1 Comparison of properties with compactness

Paracompactness is similar to compactness in the following respects:

• Every closed subset of a paracompact space is paracompact.

• Every paracompact Hausdorff space is normal.

It is different in these respects:

• A paracompact subset of a Hausdorff space need not be closed. In fact, for metric spaces, all subsets areparacompact.

• A product of paracompact spaces need not be paracompact. The square of the real line R in the lower limittopology is a classical example for this.

36.6 Variations

There are several variations of the notion of paracompactness. To define them, we first need to extend the list ofterms above:A topological space is:

• metacompact if every open cover has an open pointwise finite refinement.

• orthocompact if every open cover has an open refinement such that the intersection of all the open sets aboutany point in this refinement is open.

156 CHAPTER 36. PARACOMPACT SPACE

• fully normal if every open cover has an open star refinement, and fully T4 if it is fully normal and T1 (seeseparation axioms).

The adverb "countably" can be added to any of the adjectives “paracompact”, “metacompact”, and “fully normal” tomake the requirement apply only to countable open covers.Every paracompact space is metacompact, and every metacompact space is orthocompact.

36.6.1 Definition of relevant terms for the variations

• Given a cover and a point, the star of the point in the cover is the union of all the sets in the cover that containthe point. In symbols, the star of x in U = Uα : α in A is

U∗(x) :=∪

Uα∋x

Uα.

The notation for the star is not standardised in the literature, and this is just one possibility.

• A star refinement of a cover of a space X is a new cover of the same space such that, given any point in thespace, the star of the point in the new cover is a subset of some set in the old cover. In symbols, V is a starrefinement of U = Uα : α in A if and only if, for any x in X, there exists a Uα in U, such that V*(x) iscontained in Uα.

• A cover of a space X is pointwise finite if every point of the space belongs to only finitely many sets in thecover. In symbols, U is pointwise finite if and only if, for any x in X, the set

α ∈ A : x ∈ Uα

is finite.

As the name implies, a fully normal space is normal. Every fully T4 space is paracompact. In fact, for Hausdorffspaces, paracompactness and full normality are equivalent. Thus, a fully T4 space is the same thing as a paracompactHausdorff space.As an historical note: fully normal spaces were defined before paracompact spaces. The proof that all metrizablespaces are fully normal is easy. When it was proved by A.H. Stone that for Hausdorff spaces fully normal andparacompact are equivalent, he implicitly proved that all metrizable spaces are paracompact. Later M.E. Rudin gavea direct proof of the latter fact.

36.7 See also• a-paracompact space

• Paranormal space

36.8 Notes[1] Hatcher, Allen, Vector bundles and K-theory, preliminary version available on the author’s homepage

[2] Stone, A. H. Paracompactness and product spaces. Bull. Amer. Math. Soc. 54 (1948), 977-982

[3] Rudin, Mary Ellen. A new proof that metric spaces are paracompact. Proceedings of the American Mathematical Society,Vol. 20, No. 2. (Feb., 1969), p. 603.

[4] C. Good, I. J. Tree, and W. S. Watson. On Stone’s Theorem and the Axiom of Choice. Proceedings of the AmericanMathematical Society, Vol. 126, No. 4. (April, 1998), pp. 1211–1218.

[5] Brylinski, Jean-Luc (2007), Loop Spaces, Characteristic Classes and Geometric Quantization, Progress in Mathematics 107,Springer, p. 32, ISBN 9780817647308.

36.9. REFERENCES 157

36.9 References• Dieudonné, Jean (1944), “Une généralisation des espaces compacts”, Journal de Mathématiques Pures et Ap-pliquées, Neuvième Série 23: 65–76, ISSN 0021-7824, MR 0013297

• Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology (2 ed), Springer Verlag, 1978,ISBN 3-540-90312-7. P.23.

• Willard, Stephen (1970). General Topology. Reading, Massachusetts: Addison-Wesley. ISBN 0-486-43479-6.

• Mathew, Akhil. “Topology/Paracompactness”.

36.10 External links• Hazewinkel, Michiel, ed. (2001), “Paracompact space”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Chapter 37

Partially ordered set

x,y,z

y,zx,zx,y

y zx

Ø

The Hasse diagram of the set of all subsets of a three-element set x, y, z, ordered by inclusion. Sets on the same horizontal leveldon't share a precedence relationship. Other pairs, such as x and y,z, do not either.

In mathematics, especially order theory, a partially ordered set (or poset) formalizes and generalizes the intuitiveconcept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together witha binary relation that indicates that, for certain pairs of elements in the set, one of the elements precedes the other.Such a relation is called a partial order to reflect the fact that not every pair of elements need be related: for somepairs, it may be that neither element precedes the other in the poset. Thus, partial orders generalize the more familiartotal orders, in which every pair is related. A finite poset can be visualized through its Hasse diagram, which depictsthe ordering relation.[1]

A familiar real-life example of a partially ordered set is a collection of people ordered by genealogical descendancy.Some pairs of people bear the descendant-ancestor relationship, but other pairs bear no such relationship.

158

37.1. FORMAL DEFINITION 159

37.1 Formal definition

A (non-strict) partial order[2] is a binary relation "≤" over a set P which is reflexive, antisymmetric, and transitive,i.e., which satisfies for all a, b, and c in P:

• a ≤ a (reflexivity);

• if a ≤ b and b ≤ a then a = b (antisymmetry);

• if a ≤ b and b ≤ c then a ≤ c (transitivity).

In other words, a partial order is an antisymmetric preorder.A set with a partial order is called a partially ordered set (also called a poset). The term ordered set is sometimesalso used, as long as it is clear from the context that no other kind of order is meant. In particular, totally ordered setscan also be referred to as “ordered sets”, especially in areas where these structures are more common than posets.For a, b, elements of a partially ordered set P, if a ≤ b or b ≤ a, then a and b are comparable. Otherwise they areincomparable. In the figure on top-right, e.g. x and x,y,z are comparable, while x and y are not. A partialorder under which every pair of elements is comparable is called a total order or linear order; a totally orderedset is also called a chain (e.g., the natural numbers with their standard order). A subset of a poset in which no twodistinct elements are comparable is called an antichain (e.g. the set of singletons x, y, z in the top-rightfigure). An element a is said to be covered by another element b, written a<:b, if a is strictly less than b and no thirdelement c fits between them; formally: if both a≤b and a≠b are true, and a≤c≤b is false for each c with a≠c≠b. Amore concise definition will be given below using the strict order corresponding to "≤". For example, x is coveredby x,z in the top-right figure, but not by x,y,z.

37.2 Examples

Standard examples of posets arising in mathematics include:

• The real numbers ordered by the standard less-than-or-equal relation ≤ (a totally ordered set as well).

• The set of subsets of a given set (its power set) ordered by inclusion (see the figure on top-right). Similarly, theset of sequences ordered by subsequence, and the set of strings ordered by substring.

• The set of natural numbers equipped with the relation of divisibility.

• The vertex set of a directed acyclic graph ordered by reachability.

• The set of subspaces of a vector space ordered by inclusion.

• For a partially ordered set P, the sequence space containing all sequences of elements from P, where sequencea precedes sequence b if every item in a precedes the corresponding item in b. Formally, (an)n∈ℕ ≤ (bn) ∈ℕif and only if a ≤ b for all n in ℕ, i.e. a componentwise order.

• For a set X and a partially ordered set P, the function space containing all functions from X to P, where f ≤ gif and only if f(x) ≤ g(x) for all x in X.

• A fence, a partially ordered set defined by an alternating sequence of order relations a < b > c < d ...

37.3 Extrema

There are several notions of “greatest” and “least” element in a poset P, notably:

160 CHAPTER 37. PARTIALLY ORDERED SET

• Greatest element and least element: An element g in P is a greatest element if for every element a in P, a ≤ g.An element m in P is a least element if for every element a in P, a ≥ m. A poset can only have one greatest orleast element.

• Maximal elements and minimal elements: An element g in P is a maximal element if there is no element a inP such that a > g. Similarly, an element m in P is a minimal element if there is no element a in P such that a <m. If a poset has a greatest element, it must be the unique maximal element, but otherwise there can be morethan one maximal element, and similarly for least elements and minimal elements.

• Upper and lower bounds: For a subset A of P, an element x in P is an upper bound of A if a ≤ x, for eachelement a in A. In particular, x need not be in A to be an upper bound of A. Similarly, an element x in P is alower bound of A if a ≥ x, for each element a in A. A greatest element of P is an upper bound of P itself, anda least element is a lower bound of P.

For example, consider the positive integers, ordered by divisibility: 1 is a least element, as it divides all other elements;on the other hand this poset does not have a greatest element (although if one would include 0 in the poset, whichis a multiple of any integer, that would be a greatest element; see figure). This partially ordered set does not evenhave any maximal elements, since any g divides for instance 2g, which is distinct from it, so g is not maximal. If thenumber 1 is excluded, while keeping divisibility as ordering on the elements greater than 1, then the resulting posetdoes not have a least element, but any prime number is a minimal element for it. In this poset, 60 is an upper bound(though not a least upper bound) of the subset 2,3,5,10, which does not have any lower bound (since 1 is not in theposet); on the other hand 2 is a lower bound of the subset of powers of 2, which does not have any upper bound.

37.4 Orders on the Cartesian product of partially ordered sets

In order of increasing strength, i.e., decreasing sets of pairs, three of the possible partial orders on the Cartesianproduct of two partially ordered sets are (see figures):

• the lexicographical order: (a,b) ≤ (c,d) if a < c or (a = c and b ≤ d);

• the product order: (a,b) ≤ (c,d) if a ≤ c and b ≤ d;

• the reflexive closure of the direct product of the corresponding strict orders: (a,b) ≤ (c,d) if (a < c and b < d)or (a = c and b = d).

All three can similarly be defined for the Cartesian product of more than two sets.Applied to ordered vector spaces over the same field, the result is in each case also an ordered vector space.See also orders on the Cartesian product of totally ordered sets.

37.5 Sums of partially ordered sets

Another way to combine two posets is the ordinal sum[3] (or linear sum[4]), Z = X ⊕ Y, defined on the union of theunderlying sets X and Y by the order a ≤Z b if and only if:

• a, b ∈ X with a ≤X b, or

• a, b ∈ Y with a ≤Y b, or

• a ∈ X and b ∈ Y.

If two posets are well-ordered, then so is their ordinal sum.[5]

37.6. STRICT AND NON-STRICT PARTIAL ORDERS 161

37.6 Strict and non-strict partial orders

In some contexts, the partial order defined above is called a non-strict (or reflexive, orweak) partial order. In thesecontexts a strict (or irreflexive) partial order "<" is a binary relation that is irreflexive, transitive and asymmetric,i.e. which satisfies for all a, b, and c in P:

• not a < a (irreflexivity),

• if a < b and b < c then a < c (transitivity), and

• if a < b then not b < a (asymmetry; implied by irreflexivity and transitivity[6]).

There is a 1-to-1 correspondence between all non-strict and strict partial orders.If "≤" is a non-strict partial order, then the corresponding strict partial order "<" is the irreflexive kernel given by:

a < b if a ≤ b and a ≠ b

Conversely, if "<" is a strict partial order, then the corresponding non-strict partial order "≤" is the reflexive closuregiven by:

a ≤ b if a < b or a = b.

This is the reason for using the notation "≤".Using the strict order "<", the relation "a is covered by b" can be equivalently rephrased as "a<b, but not a<c<b forany c". Strict partial orders are useful because they correspond more directly to directed acyclic graphs (dags): everystrict partial order is a dag, and the transitive closure of a dag is both a strict partial order and also a dag itself.

37.7 Inverse and order dual

The inverse or converse ≥ of a partial order relation ≤ satisfies x≥y if and only if y≤x. The inverse of a partialorder relation is reflexive, transitive, and antisymmetric, and hence itself a partial order relation. The order dual of apartially ordered set is the same set with the partial order relation replaced by its inverse. The irreflexive relation > isto ≥ as < is to ≤.Any one of the four relations ≤, <, ≥, and > on a given set uniquely determines the other three.In general two elements x and y of a partial order may stand in any of four mutually exclusive relationships to eachother: either x < y, or x = y, or x > y, or x and y are incomparable (none of the other three). A totally ordered set is onethat rules out this fourth possibility: all pairs of elements are comparable and we then say that trichotomy holds. Thenatural numbers, the integers, the rationals, and the reals are all totally ordered by their algebraic (signed) magnitudewhereas the complex numbers are not. This is not to say that the complex numbers cannot be totally ordered; wecould for example order them lexicographically via x+iy < u+iv if and only if x < u or (x = u and y < v), but this is notordering by magnitude in any reasonable sense as it makes 1 greater than 100i. Ordering them by absolute magnitudeyields a preorder in which all pairs are comparable, but this is not a partial order since 1 and i have the same absolutemagnitude but are not equal, violating antisymmetry.

37.8 Mappings between partially ordered sets

Given two partially ordered sets (S,≤) and (T,≤), a function f: S → T is called order-preserving, or monotone,or isotone, if for all x and y in S, x≤y implies f(x) ≤ f(y). If (U,≤) is also a partially ordered set, and both f: S→ T and g: T → U are order-preserving, their composition (g∘f): S → U is order-preserving, too. A function f:S → T is called order-reflecting if for all x and y in S, f(x) ≤ f(y) implies x≤y. If f is both order-preserving andorder-reflecting, then it is called an order-embedding of (S,≤) into (T,≤). In the latter case, f is necessarily injective,since f(x) = f(y) implies x ≤ y and y ≤ x. If an order-embedding between two posets S and T exists, one says that Scan be embedded into T. If an order-embedding f: S → T is bijective, it is called an order isomorphism, and the

162 CHAPTER 37. PARTIALLY ORDERED SET

partial orders (S,≤) and (T,≤) are said to be isomorphic. Isomorphic orders have structurally similar Hasse diagrams(cf. right picture). It can be shown that if order-preserving maps f: S → T and g: T → S exist such that g∘f and f∘gyields the identity function on S and T, respectively, then S and T are order-isomorphic. [7]

For example, a mapping f: ℕ → ℙ(ℕ) from the set of natural numbers (ordered by divisibility) to the power set ofnatural numbers (ordered by set inclusion) can be defined by taking each number to the set of its prime divisors. Itis order-preserving: if x divides y, then each prime divisor of x is also a prime divisor of y. However, it is neitherinjective (since it maps both 12 and 6 to 2,3) nor order-reflecting (since besides 12 doesn't divide 6). Takinginstead each number to the set of its prime power divisors defines a map g: ℕ → ℙ(ℕ) that is order-preserving, order-reflecting, and hence an order-embedding. It is not an order-isomorphism (since it e.g. doesn't map any number tothe set 4), but it can be made one by restricting its codomain to g(ℕ). The right picture shows a subset of ℕ and itsisomorphic image under g. The construction of such an order-isomorphism into a power set can be generalized to awide class of partial orders, called distributive lattices, see "Birkhoff’s representation theorem".

37.9 Number of partial orders

Partially ordered set of set of all subsets of a six-element set a, b, c, d, e, f, ordered by the subset relation.

Sequence A001035 in OEIS gives the number of partial orders on a set of n labeled elements:The number of strict partial orders is the same as that of partial orders.If we count only up to isomorphism, we get 1, 1, 2, 5, 16, 63, 318, … (sequence A000112 in OEIS).

37.10 Linear extension

A partial order ≤* on a set X is an extension of another partial order ≤ on X provided that for all elements x and y ofX, whenever x ≤ y , it is also the case that x ≤* y. A linear extension is an extension that is also a linear (i.e., total)order. Every partial order can be extended to a total order (order-extension principle).[8]

In computer science, algorithms for finding linear extensions of partial orders (represented as the reachability ordersof directed acyclic graphs) are called topological sorting.

37.11. IN CATEGORY THEORY 163

37.11 In category theory

Every poset (and every preorder) may be considered as a category in which every hom-set has at most one element.More explicitly, let hom(x, y) = (x, y) if x ≤ y (and otherwise the empty set) and (y, z)∘(x, y) = (x, z). Posets areequivalent to one another if and only if they are isomorphic. In a poset, the smallest element, if it exists, is an initialobject, and the largest element, if it exists, is a terminal object. Also, every preordered set is equivalent to a poset.Finally, every subcategory of a poset is isomorphism-closed.

37.12 Partial orders in topological spaces

Main article: Partially ordered space

If P is a partially ordered set that has also been given the structure of a topological space, then it is customary toassume that (a, b) : a ≤ b is a closed subset of the topological product space P ×P . Under this assumption partialorder relations are well behaved at limits in the sense that if ai → a , bi → b and ai ≤ bi for all i, then a ≤ b.[9]

37.13 Interval

For a ≤ b, the closed interval [a,b] is the set of elements x satisfying a ≤ x ≤ b (i.e. a ≤ x and x ≤ b). It contains atleast the elements a and b.Using the corresponding strict relation "<", the open interval (a,b) is the set of elements x satisfying a < x < b (i.e. a< x and x < b). An open interval may be empty even if a < b. For example, the open interval (1,2) on the integers isempty since there are no integers i such that 1 < i < 2.Sometimes the definitions are extended to allow a > b, in which case the interval is empty.The half-open intervals [a,b) and (a,b] are defined similarly.A poset is locally finite if every interval is finite. For example, the integers are locally finite under their natural order-ing. The lexicographical order on the cartesian product ℕ×ℕ is not locally finite, since e.g. (1,2)≤(1,3)≤(1,4)≤(1,5)≤...≤(2,1).Using the interval notation, the property "a is covered by b" can be rephrased equivalently as [a,b] = a,b.This concept of an interval in a partial order should not be confused with the particular class of partial orders knownas the interval orders.

37.14 See also

• antimatroid, a formalization of orderings on a set that allows more general families of orderings than posets

• causal set

• comparability graph

• complete partial order

• directed set

• graded poset

• lattice

• ordered group

• poset topology, a kind of topological space that can be defined from any poset

• Scott continuity - continuity of a function between two partial orders.

• semilattice

164 CHAPTER 37. PARTIALLY ORDERED SET

• semiorder

• series-parallel partial order

• stochastic dominance

• strict weak ordering - strict partial order "<" in which the relation “neither a < b nor b < a" is transitive.

• Zorn’s lemma

37.15 Notes[1] Merrifield, Richard E.; Simmons, Howard E. (1989). Topological Methods in Chemistry. New York: John Wiley & Sons.

p. 28. ISBN 0-471-83817-9. Retrieved 27 July 2012. A partially ordered set is conveniently represented by a Hassediagram...

[2] Simovici, Dan A. & Djeraba, Chabane (2008). “Partially Ordered Sets”. Mathematical Tools for Data Mining: Set Theory,Partial Orders, Combinatorics. Springer. ISBN 9781848002012.

[3] Neggers, J.; Kim, Hee Sik (1998), “4.2 Product Order and Lexicographic Order”, Basic Posets, World Scientific, pp. 62–63,ISBN 9789810235895

[4] Davey & Priestley, Introduction to Lattices and Order (Second Edition), 2002, p. 17-18

[5] P. R. Halmos (1974). Naive Set Theory. Springer. p. 82. ISBN 978-1-4757-1645-0.

[6] Flaška, V.; Ježek, J.; Kepka, T.; Kortelainen, J. (2007). Transitive Closures of Binary Relations I (PDF). Prague: Schoolof Mathematics - Physics Charles University. p. 1. Lemma 1.1 (iv). Note that this source refers to asymmetric relationsas “strictly antisymmetric”.

[7] Davey, B. A.; Priestley, H. A. (2002). “Maps between ordered sets”. Introduction to Lattices and Order (2nd ed.). NewYork: Cambridge University Press. pp. 23–24. ISBN 0-521-78451-4. MR 1902334.

[8] Jech, Thomas (2008) [1973]. The Axiom of Choice. Dover Publications. ISBN 0-486-46624-8.

[9] Ward, L. E. Jr (1954). “Partially Ordered Topological Spaces”. Proceedings of the American Mathematical Society 5 (1):144–161. doi:10.1090/S0002-9939-1954-0063016-5

37.16 References• Deshpande, Jayant V. (1968). “On Continuity of a Partial Order”. Proceedings of the American MathematicalSociety 19 (2): 383–386. doi:10.1090/S0002-9939-1968-0236071-7.

• Schröder, Bernd S. W. (2003). Ordered Sets: An Introduction. Birkhäuser, Boston.

• Stanley, Richard P.. Enumerative Combinatorics 1. Cambridge Studies in Advanced Mathematics 49. Cam-bridge University Press. ISBN 0-521-66351-2.

37.17 External links• A001035: Number of posets with n labeled elements in the OEIS

• A000112: Number of posets with n unlabeled elements in the OEIS

Chapter 38

Partition of unity

In mathematics, a partition of unity of a topological space X is a set R of continuous functions from X to the unitinterval [0,1] such that for every point, x ∈ X ,

• there is a neighbourhood of x where all but a finite number of the functions of R are 0, and

• the sum of all the function values at x is 1, i.e.,∑

ρ∈R ρ(x) = 1 .

0

1

A partition of unity of a circle with four functions. The circle is unrolled to a line segment (the bottom solid line) for graphingpurposes. The dashed line on top is the sum of the functions in the partition.

Partitions of unity are useful because they often allow one to extend local constructions to the whole space. They arealso important in the interpolation of data, in signal processing, and the theory of spline functions.

38.1 Existence

The existence of partitions of unity assumes two distinct forms:

1. Given any open cover Uii∈I of a space, there exists a partition ρii∈I indexed over the same set I such thatsupp ρi⊆Ui. Such a partition is said to be subordinate to the open cover Uii.

2. Given any open cover Uii∈I of a space, there exists a partition ρjj∈J indexed over a possibly distinct indexset J such that each ρj has compact support and for each j∈J, supp ρj⊆Ui for some i∈I.

Thus one chooses either to have the supports indexed by the open cover, or compact supports. If the space is compact,then there exist partitions satisfying both requirements.A finite open cover always has a continuous partition of unity subordinated to it, provided the space is locally compactand Hausdorff.[1] Paracompactness of the space is a necessary condition to guarantee the existence of a partition ofunity subordinate to any open cover. Depending on the category which the space belongs to, it may also be a sufficientcondition.[2] The construction uses mollifiers (bump functions), which exist in continuous and smooth manifolds, butnot in analytic manifolds. Thus for an open cover of an analytic manifold, an analytic partition of unity subordinateto that open cover generally does not exist. See analytic continuation.If R and S are partitions of unity for spaces X and Y, respectively, then the set of all pairwise products ρσ : ρ ∈R ∧ σ ∈ S is a partition of unity for the cartesian product space X×Y.

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166 CHAPTER 38. PARTITION OF UNITY

38.2 Variant definitions

Sometimes a less restrictive definition is used: the sum of all the function values at a particular point is only requiredto be positive, rather than 1, for each point in the space. However, given such a set of functions, one can obtain apartition of unity in the strict sense by dividing every function by the sum of all functions (which is defined, since atany point it has only a finite number of terms).

38.3 Applications

A partition of unity can be used to define the integral (with respect to a volume form) of a function defined over amanifold: One first defines the integral of a function whose support is contained in a single coordinate patch of themanifold; then one uses a partition of unity to define the integral of an arbitrary function; finally one shows that thedefinition is independent of the chosen partition of unity.A partition of unity can be used to show the existence of a Riemannian metric on an arbitrary manifold.Method of steepest descent employs a partition of unity to construct asymptotics of integrals.Linkwitz–Riley filter is an example of practical implementation of partition of unity to separate input signal into twooutput signals containing only high- or low-frequency components.The Bernstein polynomials of a fixed degree m are a family of m+1 linearly independent polynomials that are apartition of unity for the unit interval [0, 1] .

38.4 See also• Gluing axiom

• Fine sheaf

38.5 References[1] Rudin, Walter (1987). Real and complex analysis (3rd ed. ed.). New York: McGraw-Hill. p. 40. ISBN 0-07-054234-1.

[2] Aliprantis, Charalambos D.; Border, Kim C. (2007). Infinite dimensional analysis: a hitchhiker’s guide (3rd ed. ed.).Berlin: Springer. p. 716. ISBN 978-3-540-32696-0.

• Tu, LoringW. (2011),An introduction to manifolds, Universitext (2nd ed.), Berlin, NewYork: Springer-Verlag,doi:10.1007/978-1-4419-7400-6, ISBN 978-1-4419-7399-3, see chapter 13

38.6 External links• General information on partition of unity at [Mathworld]

• Applications of a partition of unity at [Planet Math]

Chapter 39

Product topology

In topology and related areas of mathematics, a product space is the cartesian product of a family of topologicalspaces equipped with a natural topology called the product topology. This topology differs from another, perhapsmore obvious, topology called the box topology, which can also be given to a product space and which agrees withthe product topology when the product is over only finitely many spaces. However, the product topology is “correct”in that it makes the product space a categorical product of its factors, whereas the box topology is too fine; this is thesense in which the product topology is “natural”.

39.1 Definition

Given X such that

X :=∏i∈I

Xi

is the Cartesian product of the topological spaces Xi, indexed by i ∈ I , and the canonical projections pi : X→ Xi,the product topology on X is defined to be the coarsest topology (i.e. the topology with the fewest open sets) forwhich all the projections pi are continuous. The product topology is sometimes called the Tychonoff topology.The open sets in the product topology are unions (finite or infinite) of sets of the form

∏i∈I Ui , where each Ui is

open in Xi and Ui ≠ Xi for only finitely many i. In particular, for a finite product (in particular, for the product of twotopological spaces), the products of base elements of the Xi gives a basis for the product

∏i∈I Xi .

The product topology on X is the topology generated by sets of the form pi−1(U), where i is in I and U is an opensubset of Xi. In other words, the sets pi−1(U) form a subbase for the topology on X. A subset of X is open ifand only if it is a (possibly infinite) union of intersections of finitely many sets of the form pi−1(U). The pi−1(U) aresometimes called open cylinders, and their intersections are cylinder sets.In general, the product of the topologies of each Xi forms a basis for what is called the box topology on X. In general,the box topology is finer than the product topology, but for finite products they coincide.

39.2 Examples

If one starts with the standard topology on the real line R and defines a topology on the product of n copies of R inthis fashion, one obtains the ordinary Euclidean topology on Rn.The Cantor set is homeomorphic to the product of countably many copies of the discrete space 0,1 and the spaceof irrational numbers is homeomorphic to the product of countably many copies of the natural numbers, where againeach copy carries the discrete topology.Several additional examples are given in the article on the initial topology.

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168 CHAPTER 39. PRODUCT TOPOLOGY

39.3 Properties

The product spaceX, together with the canonical projections, can be characterized by the following universal property:IfY is a topological space, and for every i in I,fi : Y→Xi is a continuousmap, then there exists precisely one continuousmap f : Y → X such that for each i in I the following diagram commutes:

Characteristic property of product spaces

This shows that the product space is a product in the category of topological spaces. It follows from the above universalproperty that a map f : Y → X is continuous if and only if fi = pi o f is continuous for all i in I. In many cases it iseasier to check that the component functions fi are continuous. Checking whether a map f : Y → X is continuous isusually more difficult; one tries to use the fact that the pi are continuous in some way.In addition to being continuous, the canonical projections pi : X → Xi are open maps. This means that any opensubset of the product space remains open when projected down to the Xi. The converse is not true: ifW is a subspaceof the product space whose projections down to all the Xi are open, then W need not be open in X. (Consider forinstanceW = R2 \ (0,1)2.) The canonical projections are not generally closed maps (consider for example the closedset (x, y) ∈ R2 | xy = 1, whose projections onto both axes are R \ 0).The product topology is also called the topology of pointwise convergence because of the following fact: a sequence (ornet) in X converges if and only if all its projections to the spaces Xi converge. In particular, if one considers the space

39.4. RELATION TO OTHER TOPOLOGICAL NOTIONS 169

X = RI of all real valued functions on I, convergence in the product topology is the same as pointwise convergenceof functions.Any product of closed subsets of Xi is a closed set in X.An important theorem about the product topology is Tychonoff’s theorem: any product of compact spaces is compact.This is easy to show for finite products, while the general statement is equivalent to the axiom of choice.

39.4 Relation to other topological notions

• Separation

• Every product of T0 spaces is T0

• Every product of T1 spaces is T1

• Every product of Hausdorff spaces is Hausdorff[1]

• Every product of regular spaces is regular• Every product of Tychonoff spaces is Tychonoff• A product of normal spaces need not be normal

• Compactness

• Every product of compact spaces is compact (Tychonoff’s theorem)• Aproduct of locally compact spaces need not be locally compact. However, an arbitrary product of locallycompact spaces where all but finitely many are compact is locally compact (This condition is sufficientand necessary).

• Connectedness

• Every product of connected (resp. path-connected) spaces is connected (resp. path-connected)• Every product of hereditarily disconnected spaces is hereditarily disconnected.

39.5 Axiom of choice

The axiom of choice is equivalent to the statement that the product of a collection of non-empty sets is non-empty.The proof is easy enough: one needs only to pick an element from each set to find a representative in the product.Conversely, a representative of the product is a set which contains exactly one element from each component.The axiom of choice occurs again in the study of (topological) product spaces; for example, Tychonoff’s theorem oncompact sets is a more complex and subtle example of a statement that is equivalent to the axiom of choice.

39.6 See also

• Disjoint union (topology)

• Projective limit topology

• Quotient space

• Subspace (topology)

39.7 Notes[1] Product topology preserves the Hausdorff property at PlanetMath.org.

170 CHAPTER 39. PRODUCT TOPOLOGY

39.8 References• Willard, Stephen (1970). General Topology. Reading, Mass.: Addison-Wesley Pub. Co. ISBN 0486434796.Retrieved 13 February 2013.

39.9 External links• product topology at PlanetMath.org.

Chapter 40

Pseudocompact space

In mathematics, in the field of topology, a topological space is said to be pseudocompact if its image under anycontinuous function to R is bounded.

40.1 Properties related to pseudocompactness

• In order that a Tychonoff space X be pseudocompact it is necessary and sufficient that every locally finitecollection of non-empty open sets of X be finite. A series of equivalent conditions was given by Kerstan andYan-Min and other authors (see the references).

• Every countably compact space is pseudocompact. For normal Hausdorff spaces the converse is true.

• As a consequence of the above result, every sequentially compact space is pseudocompact. The converse istrue for metric spaces. As sequential compactness is an equivalent condition to compactness for metric spacesthis implies that compactness is an equivalent condition to pseudocompactness for metric spaces also.

• The weaker result that every compact space is pseudocompact is easily proved: the image of a compact spaceunder any continuous function is compact, and the Heine–Borel theorem tells us that the compact subsets of Rare precisely the closed and bounded subsets.

• If Y is the continuous image of pseudocompactX, then Y is pseudocompact. Note that for continuous functionsg : X → Y and h : Y → R, the composition of g and h, called f, is a continuous function from X to the realnumbers. Therefore, f is bounded, and Y is pseudocompact.

• Let X be an infinite set given the particular point topology. Then X is neither compact, sequentially compact,countably compact, paracompact nor metacompact. However, since X is hyperconnected, it is pseudocompact.This shows that pseudocompactness doesn't imply any other (known) form of compactness.

• In order that a Hausdorff space X be compact it is necessary and sufficient that X be pseudocompact andrealcompact (see Engelking, p. 153).

• In order that a Tychonoff space X be compact it is necessary and sufficient that X be pseudocompact andmetacompact (see Watson).

40.2 See also

• Compact space

• Paracompact space

171

172 CHAPTER 40. PSEUDOCOMPACT SPACE

• Normal space

• Realcompact space

• Metacompact space

• Tychonoff space

40.3 References• Kerstan, Johannes (1957), “Zur Charakterisierung der pseudokompakten Räume”,Mathematische Nachrichten16 (5–6): 289–293, doi:10.1002/mana.19570160505.

• W. Stephen, Watson (1981), “Pseudocompact metacompact spaces are compact”, Proc. Amer. Math. Soc. 81:151–152, doi:10.1090/s0002-9939-1981-0589159-1.

• Yan-Min, Wang (1988), “New characterisations of pseudocompact spaces”, Bull. Austral. Math. Soc. 38 (2):293–298, doi:10.1017/S0004972700027568.

• Engelking, Ryszard (1968), Outline of General Topology, translated from Polish, Amsterdam: North-Holland.

• Willard, Stephen (1970), General Topology, Reading, Mass.: Addison-Wesley.

• M.I. Voitsekhovskii (2001), “Pseudo-compact space”, in Hazewinkel, Michiel, Encyclopedia of Mathematics,Springer, ISBN 978-1-55608-010-4.

• Pseudocompact space at PlanetMath.org. .

Chapter 41

Realcompact space

In mathematics, in the field of topology, a topological space is said to be realcompact if it is completely regularHausdorff and every point of its Stone–Čech compactification is real (meaning that the quotient field at that pointof the ring of real functions is the reals). Realcompact spaces have also been called Q-spaces, saturated spaces,functionally complete spaces, real-complete spaces, replete spaces and Hewitt-Nachbin spaces (named afterEdwin Hewitt and Leopoldo Nachbin). Realcompact spaces were introduced by Hewitt (1948).

41.1 Properties

• A space is realcompact if and only if it can be embedded homeomorphically as a closed subset in some (notnecessarily finite) Cartesian power of the reals, with the product topology. Moreover, a (Hausdorff) space isrealcompact if and only if it has the uniform topology and is complete for the uniform structure generated bythe continuous real-valued functions (Gillman, Jerison, p. 226).

• For example Lindelöf spaces are realcompact; in particular all subsets of Rn are realcompact.

• The (Hewitt) realcompactification υX of a topological space X consists of the real points of its Stone–Čechcompactification βX. A topological space X is realcompact if and only if it coincides with its Hewitt realcom-pactification.

• Write C(X) for the ring of continuous functions on a topological spaceX. If Y is a real compact space, then ringhomomorphisms from C(Y) to C(X) correspond to continuous maps from X to Y. In particular the category ofrealcompact spaces is dual to the category of rings of the form C(X).

• In order that a Hausdorff space X is compact it is necessary and sufficient that X is realcompact and pseu-docompact (see Engelking, p. 153).

41.2 See also

• Compact space

• Paracompact space

• Normal space

• Pseudocompact space

• Tychonoff space

173

174 CHAPTER 41. REALCOMPACT SPACE

41.3 References• Gillman, Leonard; Jerison, Meyer, “Rings of continuous functions”. Reprint of the 1960 edition. GraduateTexts in Mathematics, No. 43. Springer-Verlag, New York-Heidelberg, 1976. xiii+300 pp.

• Hewitt, Edwin (1948), “Rings of real-valued continuous functions. I”, Transactions of the American Mathe-matical Society 64: 45–99, ISSN 0002-9947, JSTOR 1990558, MR 0026239.

• Engelking, Ryszard (1968). Outline of General Topology. translated from Polish. Amsterdam: North-HollandPubl. Co..

• Willard, Stephen (1970), General Topology, Reading, Mass.: Addison-Wesley.

Chapter 42

Regular space

In topology and related fields of mathematics, a topological space X is called a regular space if every non-emptyclosed subset C of X and a point p not contained in C admit non-overlapping open neighborhoods.[1] Thus p and Ccan be separated by neighborhoods. This condition is known as Axiom T3. The term "T3 space" usually means “aregular Hausdorff space". These conditions are examples of separation axioms.

42.1 Definitions

U

x

V

F

The point x, represented by a dot to the left of the picture, and the closed set F, represented by a closed disk to the right of the picture,are separated by their neighbourhoods U and V, represented by larger open disks. The dot x has plenty of room to wiggle aroundthe open disk U, and the closed disk F has plenty of room to wiggle around the open disk V, yet U and V do not touch each other.

A topological space X is a regular space if, given any nonempty closed set F and any point x that does not belongto F, there exists a neighbourhood U of x and a neighbourhood V of F that are disjoint. Concisely put, it must bepossible to separate x and F with disjoint neighborhoods.A T3 space or regular Hausdorff space is a topological space that is both regular and a Hausdorff space. (AHausdorff space or T2 space is a topological space in which any two distinct points are separated by neighbourhoods.)It turns out that a space is T3 if and only if it is both regular and T0. (A T0 or Kolmogorov space is a topologicalspace in which any two distinct points are topologically distinguishable, i.e., for every pair of distinct points, at leastone of them has an open neighborhood not containing the other.) Indeed, if a space is Hausdorff then it is T0, andeach T0 regular space is Hausdorff: given two distinct points, at least one of them misses the closure of the other one,so (by regularity) there exist disjoint neighborhoods separating one point from (the closure of) the other.

175

176 CHAPTER 42. REGULAR SPACE

Although the definitions presented here for “regular” and “T3" are not uncommon, there is significant variation inthe literature: some authors switch the definitions of “regular” and “T3" as they are used here, or use both termsinterchangeably. In this article, we will use the term “regular” freely, but we will usually say “regular Hausdorff”,which is unambiguous, instead of the less precise “T3". For more on this issue, see History of the separation axioms.A locally regular space is a topological space where every point has an open neighbourhood that is regular. Everyregular space is locally regular, but the converse is not true. A classical example of a locally regular space that is notregular is the bug-eyed line.

42.2 Relationships to other separation axioms

A regular space is necessarily also preregular, i.e., any two topologically distinguishable points can be separated byneighbourhoods. Since a Hausdorff space is the same as a preregular T0 space, a regular space that is also T0 must beHausdorff (and thus T3). In fact, a regular Hausdorff space satisfies the slightly stronger condition T₂½. (However,such a space need not be completely Hausdorff.) Thus, the definition of T3 may cite T0, T1, or T₂½ instead of T2

(Hausdorffness); all are equivalent in the context of regular spaces.Speaking more theoretically, the conditions of regularity and T3-ness are related by Kolmogorov quotients. A spaceis regular if and only if its Kolmogorov quotient is T3; and, as mentioned, a space is T3 if and only if it’s both regularand T0. Thus a regular space encountered in practice can usually be assumed to be T3, by replacing the space withits Kolmogorov quotient.There are many results for topological spaces that hold for both regular and Hausdorff spaces. Most of the time, theseresults hold for all preregular spaces; they were listed for regular and Hausdorff spaces separately because the ideaof preregular spaces came later. On the other hand, those results that are truly about regularity generally don't alsoapply to nonregular Hausdorff spaces.There aremany situations where another condition of topological spaces (such as normality, pseudonormality, paracompactness,or local compactness) will imply regularity if some weaker separation axiom, such as preregularity, is satisfied. Suchconditions often come in two versions: a regular version and a Hausdorff version. Although Hausdorff spaces aren'tgenerally regular, a Hausdorff space that is also (say) locally compact will be regular, because any Hausdorff space ispreregular. Thus from a certain point of view, regularity is not really the issue here, and we could impose a weakercondition instead to get the same result. However, definitions are usually still phrased in terms of regularity, sincethis condition is more well known than any weaker one.Most topological spaces studied in mathematical analysis are regular; in fact, they are usually completely regular,which is a stronger condition. Regular spaces should also be contrasted with normal spaces.

42.3 Examples and nonexamples

A zero-dimensional space with respect to the small inductive dimension has a base consisting of clopen sets. Everysuch space is regular.As described above, any completely regular space is regular, and any T0 space that is not Hausdorff (and hence notpreregular) cannot be regular. Most examples of regular and nonregular spaces studied in mathematics may be foundin those two articles. On the other hand, spaces that are regular but not completely regular, or preregular but notregular, are usually constructed only to provide counterexamples to conjectures, showing the boundaries of possibletheorems. Of course, one can easily find regular spaces that are not T0, and thus not Hausdorff, such as an indiscretespace, but these examples provide more insight on the T0 axiom than on regularity. An example of a regular spacethat is not completely regular is the Tychonoff corkscrew.Most interesting spaces in mathematics that are regular also satisfy some stronger condition. Thus, regular spaces areusually studied to find properties and theorems, such as the ones below, that are actually applied to completely regularspaces, typically in analysis.There exist Hausdorff spaces that are not regular. An example is the set R with the topology generated by sets of theform U— C, where U is an open set in the usual sense, and C is any countable subset of U.

42.4. ELEMENTARY PROPERTIES 177

42.4 Elementary properties

Suppose that X is a regular space. Then, given any point x and neighbourhood G of x, there is a closed neighbourhoodE of x that is a subset of G. In fancier terms, the closed neighbourhoods of x form a local base at x. In fact, thisproperty characterises regular spaces; if the closed neighbourhoods of each point in a topological space form a localbase at that point, then the space must be regular.Taking the interiors of these closed neighbourhoods, we see that the regular open sets form a base for the open setsof the regular space X. This property is actually weaker than regularity; a topological space whose regular open setsform a base is semiregular.

42.5 References[1] Munkres, James R. (2000). Topology (2nd ed.). Prentice Hall. ISBN 0-13-181629-2.

Chapter 43

Relatively compact subspace

In mathematics, a relatively compact subspace (or relatively compact subset) Y of a topological spaceX is a subsetwhose closure is compact.Since closed subsets of a compact space are compact, every subset of a compact space is relatively compact. In thecase of a metric topology, or more generally when sequences may be used to test for compactness, the criterion forrelative compactness becomes that any sequence in Y has a subsequence convergent in X. Such a subset may also becalled relatively bounded, or pre-compact, although the latter term is also used for a totally bounded subset. (Theseare equivalent in a complete space.)Some major theorems characterise relatively compact subsets, in particular in function spaces. An example is theArzelà–Ascoli theorem. Other cases of interest relate to uniform integrability, and the concept of normal familyin complex analysis. Mahler’s compactness theorem in the geometry of numbers characterises relatively compactsubsets in certain non-compact homogeneous spaces (specifically spaces of lattices).The definition of an almost periodic function F at a conceptual level has to do with the translates of F being a relativelycompact set. This needs to be made precise in terms of the topology used, in a particular theory.As a counterexample take any neighbourhood of the particular point of an infinite particular point space. The neigh-bourhood itself may be compact but is not relatively compact because its closure is the whole non-compact space.

43.1 See also• Compactly embedded

43.2 References• page 12 of V. Khatskevich, D.Shoikhet, Differentiable Operators and Nonlinear Equations, Birkhäuser VerlagAG, Basel, 1993, 270 pp. at google books

178

Chapter 44

Second-countable space

In topology, a second-countable space, also called a completely separable space, is a topological space satisfyingthe second axiom of countability. A space is said to be second-countable if its topology has a countable base.More explicitly, this means that a topological space T is second countable if there exists some countable collectionU = Ui∞i=1 of open subsets of T such that any open subset of T can be written as a union of elements of somesubfamily of U . Like other countability axioms, the property of being second-countable restricts the number of opensets that a space can have.Many "well-behaved" spaces in mathematics are second-countable. For example, Euclidean space (Rn) with its usualtopology is second-countable. Although the usual base of open balls is not countable, one can restrict to the set ofall open balls with rational radii and whose centers have rational coordinates. This restricted set is countable and stillforms a basis.

44.1 Properties

Second-countability is a stronger notion than first-countability. A space is first-countable if each point has a countablelocal base. Given a base for a topology and a point x, the set of all basis sets containing x forms a local base at x.Thus, if one has a countable base for a topology then one has a countable local base at every point, and hence everysecond countable space is also a first-countable space. However any uncountable discrete space is first-countable butnot second-countable.Second-countability implies certain other topological properties. Specifically, every second-countable space is separable(has a countable dense subset) and Lindelöf (every open cover has a countable subcover). The reverse implicationsdo not hold. For example, the lower limit topology on the real line is first-countable, separable, and Lindelöf, but notsecond-countable. For metric spaces, however, the properties of being second-countable, separable, and Lindelöf areall equivalent. Therefore, the lower limit topology on the real line is not metrizable.In second-countable spaces—as in metric spaces—compactness, sequential compactness, and countable compactnessare all equivalent properties.Urysohn's metrization theorem states that every second-countable, regular space is metrizable. It follows that everysuch space is completely normal as well as paracompact. Second-countability is therefore a rather restrictive propertyon a topological space, requiring only a separation axiom to imply metrizability.

44.1.1 Other properties

• A continuous, open image of a second-countable space is second-countable.

• Every subspace of a second-countable space is second-countable.

• Quotients of second-countable spaces need not be second-countable; however, open quotients always are.

• Any countable product of a second-countable space is second-countable, although uncountable products neednot be.

179

180 CHAPTER 44. SECOND-COUNTABLE SPACE

• The topology of a second-countable space has cardinality less than or equal to c (the cardinality of the contin-uum).

• Any base for a second-countable space has a countable subfamily which is still a base.

• Every collection of disjoint open sets in a second-countable space is countable.

44.2 Examples• Consider the disjoint countable union X = [0, 1] ∪ [2, 3] ∪ [4, 5] ∪ · · · ∪ [2k, 2k + 1] ∪ · · · . Define anequivalence relation and a quotient topology by identifying the left ends of the intervals - that is, identify 0 ~ 2~ 4 ~ … ~ 2k and so on. X is second countable, as a countable union of second countable spaces. However,X/~ is not first countable at the coset of the identified points and hence also not second countable.

• Note that the above space is not homeomorphic to the same set of equivalence classes endowedwith the obviousmetric: i.e. regular Euclidean distance for two points in the same interval, and the sum of the distances to theleft hand point for points not in the same interval. It is a separable metric space (consider the set of rationalpoints), and hence is second-countable.

• The long line is not second countable.

44.3 References• Stephen Willard, General Topology, (1970) Addison-Wesley Publishing Company, Reading Massachusetts.

• John G. Hocking and Gail S. Young (1961). Topology. Corrected reprint, Dover, 1988. ISBN 0-486-65676-4

Chapter 45

Sequence

“Sequential” redirects here. For the manual transmission, see Sequential manual transmission. For other uses, seeSequence (disambiguation).

In mathematics, a sequence is an ordered collection of objects in which repetitions are allowed. Like a set, it containsmembers (also called elements, or terms). The number of elements (possibly infinite) is called the length of thesequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positionsin the sequence. Formally, a sequence can be defined as a function whose domain is a countable totally ordered set,such as the natural numbers.For example, (M, A, R, Y) is a sequence of letters with the letter 'M' first and 'Y' last. This sequence differs from(A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is avalid sequence. Sequences can be finite, as in these examples, or infinite, such as the sequence of all even positiveintegers (2, 4, 6,...). In computing and computer science, finite sequences are sometimes called strings, words or lists,the different names commonly corresponding to different ways to represent them into computer memory; infinitesequences are also called streams. The empty sequence ( ) is included in most notions of sequence, but may beexcluded depending on the context.

An infinite sequence of real numbers (in blue). This sequence is neither increasing, nor decreasing, nor convergent, nor Cauchy. Itis, however, bounded.

181

182 CHAPTER 45. SEQUENCE

45.1 Examples and notation

A sequence can be thought of as a list of elements with a particular order. Sequences are useful in a number ofmathematical disciplines for studying functions, spaces, and other mathematical structures using the convergenceproperties of sequences. In particular, sequences are the basis for series, which are important in differential equationsand analysis. Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in thestudy of prime numbers.There are a number of ways to denote a sequence, some of which are more useful for specific types of sequences.One way to specify a sequence is to list the elements. For example, the first four odd numbers form the sequence(1,3,5,7). This notation can be used for infinite sequences as well. For instance, the infinite sequence of positiveodd integers can be written (1,3,5,7,...). Listing is most useful for infinite sequences with a pattern that can be easilydiscerned from the first few elements. Other ways to denote a sequence are discussed after the examples.

45.1.1 Important examples

3 2

1 1

5

8

A tiling with squares whose sides are successive Fibonacci numbers in length.

There are many important integer sequences. The prime numbers are the natural numbers bigger than 1, that haveno divisors but 1 and themselves. Taking these in their natural order gives the sequence (2,3,5,7,11,13,17,...). Thestudy of prime numbers has important applications for mathematics and specifically number theory.The Fibonacci numbers are the integer sequence whose elements are the sum of the previous two elements. The firsttwo elements are either 0 and 1 or 1 and 1 so that the sequence is (0,1,1,2,3,5,8,13,21,34,...).Other interesting sequences include the ban numbers, whose spellings do not contain a certain letter of the alphabet.For instance, the eban numbers (do not contain 'e') form the sequence (2,4,6,30,32,34,36,40,42,...). Another sequencebased on the English spelling of the letters is the one based on their number of letters (3,3,5,4,4,3,5,5,4,3,6,6,8,...).For a list of important examples of integers sequences see On-line Encyclopedia of Integer Sequences.Other important examples of sequences include ones made up of rational numbers, real numbers, and complex num-bers. The sequence (.9,.99,.999,.9999,...) approaches the number 1. In fact, every real number can be written asthe limit of a sequence of rational numbers. It is this fact that allows us to write any real number as the limit of asequence of decimals. For instance, π is the limit of the sequence (3,3.1,3.14,3.141,3.1415,...). The sequence for π,however, does not have any pattern that is easily discernible by eye, unlike the sequence (0.9,0.99,...).

45.1. EXAMPLES AND NOTATION 183

45.1.2 Indexing

Other notations can be useful for sequences whose pattern cannot be easily guessed, or for sequences that do not havea pattern such as the digits of π. This section focuses on the notations used for sequences that are a map from a subsetof the natural numbers. For generalizations to other countable index sets see the following section and below.The terms of a sequence are commonly denoted by a single variable, say an, where the index n indicates the nthelement of the sequence.

a1 ↔ element 1sta2 ↔ element 2nda3 ↔ element 3rd...

...an−1 ↔ element (n-1)than ↔ element nth

an+1 ↔ element (n+1)th...

...

Indexing notation is used to refer to a sequence in the abstract. It is also a natural notation for sequences whoseelements are related to the index n (the element’s position) in a simple way. For instance, the sequence of the first 10square numbers could be written as

(a1, a2, ..., a10), ak = k2.

This represents the sequence (1,4,9,...100). This notation is often simplified further as

(ak)10k=1, ak = k2.

Here the subscript k=1 and superscript 10 together tell us that the elements of this sequence are the ak such that k= 1, 2, ..., 10.Sequences can be indexed beginning and ending from any integer. The infinity symbol ∞ is often used as the super-script to indicate the sequence including all integer k-values starting with a certain one. The sequence of all positivesquares is then denoted

(ak)∞k=1, ak = k2.

In cases where the set of indexing numbers is understood, such as in analysis, the subscripts and superscripts are oftenleft off. That is, one simply writes ak for an arbitrary sequence. In analysis, k would be understood to run from 1 to∞. However, sequences are often indexed starting from zero, as in

(ak)∞k=0 = (a0, a1, a2, ...).

In some cases the elements of the sequence are related naturally to a sequence of integers whose pattern can be easilyinferred. In these cases the index set may be implied by a listing of the first few abstract elements. For instance, thesequence of squares of odd numbers could be denoted in any of the following ways.

• (1, 9, 25, ...)

• (a1, a3, a5, ...), ak = k2

• (a2k−1)∞k=1, ak = k2

184 CHAPTER 45. SEQUENCE

• (ak)∞k=1, ak = (2k − 1)2

• ((2k − 1)2)∞k=1

Moreover, the subscripts and superscripts could have been left off in the third, fourth, and fifth notations if theindexing set was understood to be the natural numbers.Finally, sequences can more generally be denoted by writing a set inclusion in the subscript, such as in

(ak)k∈N

The set of values that the index can take on is called the index set. In general, the ordering of the elements ak isspecified by the order of the elements in the indexing set. When N is the index set, the element ak+1 comes after theelement ak since in N, the element (k+1) comes directly after the element k.

45.1.3 Specifying a sequence by recursion

Sequences whose elements are related to the previous elements in a straightforward way are often specified usingrecursion. This is in contrast to the specification of sequence elements in terms of their position.To specify a sequence by recursion requires a rule to construct each consecutive element in terms of the ones beforeit. In addition, enough initial elements must be specified so that new elements of the sequence can be specified bythe rule. The principle of mathematical induction can be used to prove that a sequence is well-defined, which is tosay that that every element of the sequence is specified at least once and has a single, unambiguous value. Inductioncan also be used to prove properties about a sequence, especially for sequences whose most natural specification isby recursion.The Fibonacci sequence can be defined using a recursive rule along with two initial elements. The rule is that eachelement is the sum of the previous two elements, and the first two elements are 0 and 1.

an = an−1 + an−2 , with a0 = 0 and a1 = 1.

The first ten terms of this sequence are 0,1,1,2,3,5,8,13,21, and 34. A more complicated example of a sequence thatis defined recursively is Recaman’s sequence, considered at the beginning of this section. We can define Recaman’ssequence by

a0 = 0 and an = an−1−n if the result is positive and not already in the list. Otherwise, an = an−1+n .

Not all sequences can be specified by a rule in the form of an equation, recursive or not, and some can be quitecomplicated. For example, the sequence of prime numbers is the set of prime numbers in their natural order. Thisgives the sequence (2,3,5,7,11,13,17,...).One can also notice that the next element of a sequence is a function of the element before, and so we can write thenext element as : an+1 = f(an)

This functional notation can prove useful when one wants to prove the global monotony of the sequence.

45.2 Formal definition and basic properties

There are many different notions of sequences in mathematics, some of which (e.g., exact sequence) are not coveredby the definitions and notations introduced below.

45.2.1 Formal definition

A sequence is usually defined as a function whose domain is a countable totally ordered set, although in many disci-plines the domain is restricted, such as to the natural numbers. In real analysis a sequence is a function from a subset

45.2. FORMAL DEFINITION AND BASIC PROPERTIES 185

of the natural numbers to the real numbers.[1] In other words, a sequence is a map f(n) : N → R. To recover ourearlier notation we might identify an = f(n) for all n or just write an : N→ R.In complex analysis, sequences are defined as maps from the natural numbers to the complex numbers (C).[2] Intopology, sequences are often defined as functions from a subset of the natural numbers to a topological space.[3]Sequences are an important concept for studying functions and, in topology, topological spaces. An important gener-alization of sequences, called a net, is to functions from a (possibly uncountable) directed set to a topological space.

45.2.2 Finite and infinite

The length of a sequence is defined as the number of terms in the sequence.A sequence of a finite length n is also called an n-tuple. Finite sequences include the empty sequence ( ) that has noelements.Normally, the term infinite sequence refers to a sequence which is infinite in one direction, and finite in the other—thesequence has a first element, but no final element, it is called a singly infinite, or one-sided (infinite) sequence,when disambiguation is necessary. In contrast, a sequence that is infinite in both directions—i.e. that has neither afirst nor a final element—is called a bi-infinite sequence, two-way infinite sequence, or doubly infinite sequence.A function from the set Z of all integers into a set, such as for instance the sequence of all even integers ( …, −4, −2,0, 2, 4, 6, 8… ), is bi-infinite. This sequence could be denoted (2n)∞n=−∞ .One can interpret singly infinite sequences as elements of the semigroup ring of the natural numbers R[N], and doublyinfinite sequences as elements of the group ring of the integers R[Z]. This perspective is used in the Cauchy productof sequences.

45.2.3 Increasing and decreasing

A sequence is said to be monotonically increasing if each term is greater than or equal to the one before it. For asequence (an)∞n=1 this can be written as an ≤ an₊₁ for all n ∈ N. If each consecutive term is strictly greater than(>) the previous term then the sequence is called strictly monotonically increasing. A sequence is monotonicallydecreasing if each consecutive term is less than or equal to the previous one, and strictly monotonically decreasingif each is strictly less than the previous. If a sequence is either increasing or decreasing it is called a monotonesequence. This is a special case of the more general notion of a monotonic function.The terms nondecreasing and nonincreasing are often used in place of increasing and decreasing in order to avoidany possible confusion with strictly increasing and strictly decreasing, respectively.

45.2.4 Bounded

If the sequence of real numbers (an) is such that all the terms, after a certain one, are less than some real numberM,then the sequence is said to be bounded from above. In less words, this means an ≤ M for all n greater than N forsome pair M and N. Any such M is called an upper bound. Likewise, if, for some real m, an ≥ m for all n greaterthan some N, then the sequence is bounded from below and any such m is called a lower bound. If a sequence isboth bounded from above and bounded from below then the sequence is said to be bounded.

45.2.5 Other types of sequences

A subsequence of a given sequence is a sequence formed from the given sequence by deleting some of the elementswithout disturbing the relative positions of the remaining elements. For instance, the sequence of positive evenintegers (2,4,6,...) is a subsequence of the positive integers (1,2,3,...). The positions of some elements change whenother elements are deleted. However, the relative positions are preserved.Some other types of sequences that are easy to define include:

• An integer sequence is a sequence whose terms are integers.

• A polynomial sequence is a sequence whose terms are polynomials.

186 CHAPTER 45. SEQUENCE

• A positive integer sequence is sometimes calledmultiplicative if anm = an am for all pairs n,m such that n andm are coprime.[4] In other instances, sequences are often called multiplicative if an = na1 for all n. Moreover,the multiplicative Fibonacci sequence satisfies the recursion relation an = an₋₁ an₋₂.

45.3 Limits and convergence

Main article: Limit of a sequenceOne of the most important properties of a sequence is convergence. Informally, a sequence converges if it has a limit.

5 10 15 20 25

0.0

0.2

0.4

0.6

0.8

1.0

n + 12n2

The plot of a convergent sequence (a ) is shown in blue. Visually we can see that the sequence is converging to the limit zero as nincreases.

Continuing informally, a (singly infinite) sequence has a limit if it approaches some value L, called the limit, as nbecomes very large. That is, for an abstract sequence (an) (with n running from 1 to infinity understood) the valueof an approaches L as n→∞, denoted

limn→∞

an = L.

More precisely, the sequence converges if there exists a limit L such that the remaining a 's are arbitrarily close to Lfor some n large enough.If a sequence converges to some limit, then it is convergent; otherwise it is divergent.If an gets arbitrarily large as n→∞ we write

limn→∞

an = ∞.

45.3. LIMITS AND CONVERGENCE 187

In this case we say that the sequence (an) diverges, or that it converges to infinity.If an becomes arbitrarily “small” negative numbers (large in magnitude) as n→∞ we write

limn→∞

an = −∞

and say that the sequence diverges or converges to minus infinity.

45.3.1 Definition of convergence

For sequences that can be written as (an)∞n=1 with an ∈ R we can write (an) with the indexing set understood as N.These sequences are most common in real analysis. The generalizations to other types of sequences are consideredin the following section and the main page Limit of a sequence.Let (an) be a sequence. In words, the sequence (an) is said to converge if there exists a number L such that no matterhow close we want the an to be to L (say ε-close where ε > 0), we can find a natural number N such that all terms(aN+1, aN+2, ...) are further closer to L (within ε of L). [1] This is often written more compactly using symbols. Forinstance,

for all ε > 0, there exists a natural number N such that L−ε < an < L+ε for all n ≥ N.

In even more compact notation

∀ϵ > 0, ∃N ∈ N s.t. ∀n ≥ N, |an − L| < ϵ.

The difference in the definitions of convergence for (one-sided) sequences in complex analysis and metric spaces isthat the absolute value |an − L| is interpreted as the distance in the complex plane (

√z∗z ), and the distance under

the appropriate metric, respectively.

45.3.2 Applications and important results

Important results for convergence and limits of (one-sided) sequences of real numbers include the following. Theseequalities are all true at least when both sides exist. For a discussion of when the existence of the limit on one sideimplies the existence of the other see a real analysis text such as can be found in the references.[1][5]

• The limit of a sequence is unique.

• limn→∞(an ± bn) = limn→∞ an ± limn→∞ bn

• limn→∞ can = c limn→∞ an

• limn→∞(anbn) = (limn→∞ an)(limn→∞ bn)

• limn→∞an

bn= limn→∞ an

limn→∞ bnprovided limn→∞ bn = 0

• limn→∞ apn = [limn→∞ an]p

• If an ≤ bn for all n greater than some N, then limn→∞ an ≤ limn→∞ bn .

• (Squeeze Theorem) If an ≤ cn ≤ bn for all n >N, and limn→∞ an = limn→∞ bn = L , then limn→∞ cn = L.

• If a sequence is bounded and monotonic then it is convergent.

• A sequence is convergent if and only if every subsequence is convergent.

188 CHAPTER 45. SEQUENCE

The plot of a Cauchy sequence (X ), shown in blue, asX versus n. Visually, we see that the sequence appears to be converging to thelimit zero as the terms in the sequence become closer together as n increases. In the real numbers every Cauchy sequence convergesto some limit.

45.3.3 Cauchy sequences

Main article: Cauchy sequenceA Cauchy sequence is a sequence whose terms become arbitrarily close together as n gets very large. The notion ofa Cauchy sequence is important in the study of sequences in metric spaces, and, in particular, in real analysis. Oneparticularly important result in real analysis is Cauchy characterization of convergence for sequences:

In the real numbers, a sequence is convergent if and only if it is Cauchy.

In contrast, in the rational numbers, e.g. the sequence defined by x1 = 1 and xn₊₁ = xn + 2/xn/2 is Cauchy, but has norational limit, cf. here.

45.4 Series

Main article: Series (mathematics)

A series is, informally speaking, the sum of the terms of a sequence. That is, adding the first N terms of a (one-sided)sequence forms the Nth term of another sequence, called a series. Thus the N series of the sequence (a ) results inanother sequence (SN) given by:

S1 = a1

S2 = a1 + a2

S3 = a1 + a2 + a3

......

SN = a1 + a2 + a3 + · · ·...

...

We can also write the nth term of the series as

45.5. USE IN OTHER FIELDS OF MATHEMATICS 189

SN =N∑

n=1

an.

Then the concepts used to talk about sequences, such as convergence, carry over to series (the sequence of partialsums) and the properties can be characterized as properties of the underlying sequences (such as (an) in the lastexample). The limit, if it exists, of an infinite series (the series created from an infinite sequence) is written as

limN→∞

SN =

∞∑n=1

an.

45.5 Use in other fields of mathematics

45.5.1 Topology

Sequence play an important role in topology, especially in the study of metric spaces. For instance:

• A metric space is compact exactly when it is sequentially compact.

• A function from ametric space to another metric space is continuous exactly when it takes convergent sequencesto convergent sequences.

• A metric space is a connected space if, whenever the space is partitioned into two sets, one of the two setscontains a sequence converging to a point in the other set.

• A topological space is separable exactly when there is a dense sequence of points.

Sequences can be generalized to nets or filters. These generalizations allow one to extend some of the above theoremsto spaces without metrics.

Product topology

A product space of a sequence of topological spaces is the cartesian product of the spaces equipped with a naturaltopology called the product topology.More formally, given a sequence of spaces Xi , define X such that

X :=∏i∈I

Xi,

is the set of sequences xi where each xi is an element of Xi . Let the canonical projections be written as pi :X → Xi. Then the product topology on X is defined to be the coarsest topology (i.e. the topology with the fewestopen sets) for which all the projections pi are continuous. The product topology is sometimes called the Tychonofftopology.

45.5.2 Analysis

In analysis, when talking about sequences, one will generally consider sequences of the form

(x1, x2, x3, . . . ) or (x0, x1, x2, . . . )

which is to say, infinite sequences of elements indexed by natural numbers.

190 CHAPTER 45. SEQUENCE

It may be convenient to have the sequence start with an index different from 1 or 0. For example, the sequence definedby xn = 1/log(n) would be defined only for n ≥ 2. When talking about such infinite sequences, it is usually sufficient(and does not change much for most considerations) to assume that the members of the sequence are defined at leastfor all indices large enough, that is, greater than some given N.The most elementary type of sequences are numerical ones, that is, sequences of real or complex numbers. This typecan be generalized to sequences of elements of some vector space. In analysis, the vector spaces considered are oftenfunction spaces. Even more generally, one can study sequences with elements in some topological space.

Sequence spaces

Main article: Sequence space

A sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, itis a function space whose elements are functions from the natural numbers to the fieldK of real or complex numbers.The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in K,and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalarmultiplication. All sequence spaces are linear subspaces of this space. Sequence spaces are typically equipped witha norm, or at least the structure of a topological vector space.The most important sequences spaces in analysis are the ℓp spaces, consisting of the p-power summable sequences,with the p-norm. These are special cases of Lp spaces for the counting measure on the set of natural numbers.Other important classes of sequences like convergent sequences or null sequences form sequence spaces, respectivelydenoted c and c0, with the sup norm. Any sequence space can also be equipped with the topology of pointwiseconvergence, under which it becomes a special kind of Fréchet space called FK-space.

45.5.3 Linear algebra

Sequences over a field may also be viewed as vectors in a vector space. Specifically, the set of F-valued sequences(where F is a field) is a function space (in fact, a product space) of F-valued functions over the set of natural numbers.

45.5.4 Abstract algebra

Abstract algebra employs several types of sequences, including sequences of mathematical objects such as groups orrings.

Free monoid

Main article: Free monoid

If A is a set, the free monoid over A (denoted A*, also called Kleene star of A) is a monoid containing all the finitesequences (or strings) of zero or more elements of A, with the binary operation of concatenation. The free semigroupA+ is the subsemigroup of A* containing all elements except the empty sequence.

Exact sequences

Main article: Exact sequence

In the context of group theory, a sequence

G0f1−→ G1

f2−→ G2f3−→ · · · fn−→ Gn

of groups and group homomorphisms is called exact if the image (or range) of each homomorphism is equal to thekernel of the next:

45.6. TYPES 191

im(fk) = ker(fk+1)

Note that the sequence of groups and homomorphisms may be either finite or infinite.A similar definition can be made for certain other algebraic structures. For example, one could have an exact sequenceof vector spaces and linear maps, or of modules and module homomorphisms.

Spectral sequences

Main article: Spectral sequence

In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groupsby taking successive approximations. Spectral sequences are a generalization of exact sequences, and since theirintroduction by Jean Leray (1946), they have become an important research tool, particularly in homotopy theory.

45.5.5 Set theory

An ordinal-indexed sequence is a generalization of a sequence. If α is a limit ordinal and X is a set, an α-indexedsequence of elements of X is a function from α to X. In this terminology an ω-indexed sequence is an ordinarysequence.

45.5.6 Computing

Automata or finite statemachines can typically be thought of as directed graphs, with edges labeled using some specificalphabet, Σ. Most familiar types of automata transition from state to state by reading input letters from Σ, followingedges with matching labels; the ordered input for such an automaton forms a sequence called a word (or input word).The sequence of states encountered by the automaton when processing a word is called a run. A nondeterministicautomaton may have unlabeled or duplicate out-edges for any state, giving more than one successor for some inputletter. This is typically thought of as producing multiple possible runs for a given word, each being a sequence ofsingle states, rather than producing a single run that is a sequence of sets of states; however, 'run' is occasionally usedto mean the latter.

45.5.7 Streams

Infinite sequences of digits (or characters) drawn from a finite alphabet are of particular interest in theoretical com-puter science. They are often referred to simply as sequences or streams, as opposed to finite strings. Infinite binarysequences, for instance, are infinite sequences of bits (characters drawn from the alphabet 0, 1). The set C = 0,1∞ of all infinite, binary sequences is sometimes called the Cantor space.An infinite binary sequence can represent a formal language (a set of strings) by setting the n th bit of the sequence to1 if and only if the n th string (in shortlex order) is in the language. This representation is useful in the diagonalizationmethod for proofs.[6]

45.6 Types

• ±1-sequence

• Arithmetic progression

• Cauchy sequence

• Farey sequence

• Fibonacci sequence

192 CHAPTER 45. SEQUENCE

• Geometric progression

• Look-and-say sequence

• Thue–Morse sequence

45.7 Related concepts

• List (computing)

• Ordinal-indexed sequence

• Recursion (computer science)

• Tuple

• Set theory

45.8 Operations

• Cauchy product

• Limit of a sequence

45.9 See also

• Enumeration

• Net (topology) (a generalization of sequences)

• On-Line Encyclopedia of Integer Sequences

• Permutation

• Recurrence relation

• Sequence space

• Set (mathematics)

45.10 References

[1] Gaughan, Edward. “1.1 Sequences and Convergence”. Introduction to Analysis. AMS (2009). ISBN 0-8218-4787-2.

[2] Edward B. Saff & Arthur David Snider (2003). “Chapter 2.1”. Fundamentals of Complex Analysis. ISBN 01-390-7874-6.

[3] James R. Munkres. “Chapters 1&2”. Topology. ISBN 01-318-1629-2.

[4] Lando, Sergei K. “7.4 Multiplicative sequences”. Lectures on generating functions. AMS. ISBN 0-8218-3481-9.

[5] Dawikins, Paul. “Series and Sequences”. Paul’s Online Math Notes/Calc II (notes). Retrieved 18 December 2012.

[6] Oflazer, Kemal. “FORMAL LANGUAGES, AUTOMATAANDCOMPUTATION: DECIDABILITY” (PDF). cmu.edu.Carnegie-Mellon University. Retrieved 24 April 2015.

45.11. EXTERNAL LINKS 193

45.11 External links• The dictionary definition of sequence at Wiktionary

• Hazewinkel, Michiel, ed. (2001), “Sequence”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• The On-Line Encyclopedia of Integer Sequences

• Journal of Integer Sequences (free)

• Sequence at PlanetMath.org.

Chapter 46

Sequentially compact space

In mathematics, a topological space is sequentially compact if every infinite sequence has a convergent subsequence.For general topological spaces, the notions of compactness and sequential compactness are not equivalent; they are,however, equivalent for metric spaces. A metric space X is sequentially compact if every sequence has a convergentsubsequence which converges to a point in X.

46.1 Examples and properties

The space of all real numbers with the standard topology is not sequentially compact; the sequence (sn = n) for allnatural numbers n is a sequence that has no convergent subsequence.If a space is a metric space, then it is sequentially compact if and only if it is compact.[1] However in general thereexist sequentially compact spaces that are not compact (such as the first uncountable ordinal with the order topology),and compact spaces that are not sequentially compact (such as the product of 2ℵ0 = c copies of the closed unitinterval).[2]

46.2 Related notions

• A topological space X is said to be limit point compact if every infinite subset of X has a limit point in X.

• A topological space is countably compact if every countable open cover has a finite subcover.

In a metric space, the notions of sequential compactness, limit point compactness, countable compactness andcompactness are equivalent.In a sequential space sequential compactness is equivalent to countable compactness.[3]

There is also a notion of a one-point sequential compactification -- the idea is that the non convergent sequencesshould all converge to the extra point. See [4]

46.3 See also

• Bolzano–Weierstrass theorem

46.4 Notes[1] Willard, 17G, p. 125.

[2] Steen and Seebach, Example 105, pp. 125—126.

194

46.5. REFERENCES 195

[3] Engelking, General Topology, Theorem 3.10.31K.P. Hart, Jun-iti Nagata, J.E. Vaughan (editors), Encyclopedia of General Topology, Chapter d3 (by P. Simon)

[4] Brown, Ronald, “Sequentially proper maps and a sequential compactification”, J. London Math Soc. (2) 7 (1973) 515-522.

46.5 References• Munkres, James (1999). Topology (2nd edition ed.). Prentice Hall. ISBN 0-13-181629-2.

• Steen, Lynn A. and Seebach, J. Arthur Jr.; Counterexamples in Topology, Holt, Rinehart and Winston (1970).ISBN 0-03-079485-4.

• Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.

Chapter 47

Set (mathematics)

This article is about what mathematicians call “intuitive” or “naive” set theory. For a more detailed account, see Naiveset theory. For a rigorous modern axiomatic treatment of sets, see Set theory.In mathematics, a set is a collection of distinct objects, considered as an object in its own right. For example,

A set of polygons in a Venn diagram

the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectivelythey form a single set of size three, written 2,4,6. Sets are one of the most fundamental concepts in mathematics.Developed at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as afoundation from which nearly all of mathematics can be derived. In mathematics education, elementary topics suchas Venn diagrams are taught at a young age, while more advanced concepts are taught as part of a university degree.

196

47.1. DEFINITION 197

The German word Menge, rendered as “set” in English, was coined by Bernard Bolzano in his work The Paradoxesof the Infinite.

47.1 Definition

Passage with the original set definition of Georg Cantor

A set is a well defined collection of distinct objects. The objects that make up a set (also known as the elements ormembers of a set) can be anything: numbers, people, letters of the alphabet, other sets, and so on. Georg Cantor,the founder of set theory, gave the following definition of a set at the beginning of his Beiträge zur Begründung dertransfiniten Mengenlehre:[1]

A set is a gathering together into a whole of definite, distinct objects of our perception [Anschauung]or of our thought—which are called elements of the set.

Sets are conventionally denoted with capital letters. Sets A and B are equal if and only if they have precisely the sameelements.[2]

There is the image popular, that sets are like boxes containing their elements. But there is a huge difference betweenboxes and sets. While boxes don't change their identity when objects are removed from or added to them, sets changetheir identity when their elements change. So its better to have the image of a set as the content of an imaginary box:

• A set of polygons

• The same set as a box

• The set as the content of a box

Cantor’s definition turned out to be inadequate for formal mathematics; instead, the notion of a “set” is taken as anundefined primitive in axiomatic set theory, and its properties are defined by the Zermelo–Fraenkel axioms. Themost basic properties are that a set has elements, and that two sets are equal (one and the same) if and only if everyelement of each set is an element of the other.

47.2 Describing sets

Main article: Set notation

There are two ways of describing, or specifying the members of, a set. One way is by intensional definition, using arule or semantic description:

198 CHAPTER 47. SET (MATHEMATICS)

A is the set whose members are the first four positive integers.B is the set of colors of the French flag.

The second way is by extension – that is, listing each member of the set. An extensional definition is denoted byenclosing the list of members in curly brackets:

C = 4, 2, 1, 3D = blue, white, red.

One often has the choice of specifying a set either intensionally or extensionally. In the examples above, for instance,A = C and B = D.There are two important points to note about sets. First, a set can have two or more members which are identical, forexample, 11, 6, 6. However, we say that two sets which differ only in that one has duplicate members are in factexactly identical (see Axiom of extensionality). Hence, the set 11, 6, 6 is exactly identical to the set 11, 6. Thesecond important point is that the order in which the elements of a set are listed is irrelevant (unlike for a sequenceor tuple). We can illustrate these two important points with an example:

6, 11 = 11, 6 = 11, 6, 6, 11 .

For sets with many elements, the enumeration of members can be abbreviated. For instance, the set of the firstthousand positive integers may be specified extensionally as

1, 2, 3, ..., 1000,

where the ellipsis ("...”) indicates that the list continues in the obvious way. Ellipses may also be used where sets haveinfinitely many members. Thus the set of positive even numbers can be written as 2, 4, 6, 8, ... .The notation with braces may also be used in an intensional specification of a set. In this usage, the braces have themeaning “the set of all ...”. So, E = playing card suits is the set whose four members are ♠, ♦, ♥, and ♣. A moregeneral form of this is set-builder notation, through which, for instance, the set F of the twenty smallest integers thatare four less than perfect squares can be denoted

F = n2 − 4 : n is an integer; and 0 ≤ n ≤ 19.

In this notation, the colon (":") means “such that”, and the description can be interpreted as "F is the set of all numbersof the form n2 − 4, such that n is a whole number in the range from 0 to 19 inclusive.” Sometimes the vertical bar("|") is used instead of the colon.

47.3 Membership

Main article: Element (mathematics)

If B is a set and x is one of the objects of B, this is denoted x ∈ B, and is read as “x belongs to B”, or “x is an elementof B”. If y is not a member of B then this is written as y ∉ B, and is read as “y does not belong to B”.For example, with respect to the sets A = 1,2,3,4, B = blue, white, red, and F = n2 − 4 : n is an integer; and 0≤ n ≤ 19 defined above,

4 ∈ A and 12 ∈ F; but9 ∉ F and green ∉ B.

47.3. MEMBERSHIP 199

47.3.1 Subsets

Main article: Subset

If every member of set A is also a member of set B, then A is said to be a subset of B, written A ⊆ B (also pronouncedA is contained in B). Equivalently, we can write B ⊇ A, read as B is a superset of A, B includes A, or B contains A. Therelationship between sets established by ⊆ is called inclusion or containment.If A is a subset of, but not equal to, B, then A is called a proper subset of B, written A ⊊ B (A is a proper subset of B)or B ⊋ A (B is a proper superset of A).Note that the expressions A ⊂ B and B ⊃ A are used differently by different authors; some authors use them to meanthe same as A ⊆ B (respectively B ⊇ A), whereas other use them to mean the same as A ⊊ B (respectively B ⊋ A).

AB

A is a subset of B

Example:

• The set of all men is a proper subset of the set of all people.• 1, 3 ⊆ 1, 2, 3, 4.• 1, 2, 3, 4 ⊆ 1, 2, 3, 4.

200 CHAPTER 47. SET (MATHEMATICS)

The empty set is a subset of every set and every set is a subset of itself:

• ∅ ⊆ A.• A ⊆ A.

An obvious but useful identity, which can often be used to show that two seemingly different sets are equal:

• A = B if and only if A ⊆ B and B ⊆ A.

A partition of a set S is a set of nonempty subsets of S such that every element x in S is in exactly one of these subsets.

47.3.2 Power sets

Main article: Power set

The power set of a set S is the set of all subsets of S. Note that the power set contains S itself and the empty setbecause these are both subsets of S. For example, the power set of the set 1, 2, 3 is 1, 2, 3, 1, 2, 1, 3, 2,3, 1, 2, 3, ∅. The power set of a set S is usually written as P(S).The power set of a finite set with n elements has 2n elements. This relationship is one of the reasons for the terminologypower set. For example, the set 1, 2, 3 contains three elements, and the power set shown above contains 23 = 8elements.The power set of an infinite (either countable or uncountable) set is always uncountable. Moreover, the power set ofa set is always strictly “bigger” than the original set in the sense that there is no way to pair every element of S withexactly one element of P(S). (There is never an onto map or surjection from S onto P(S).)Every partition of a set S is a subset of the powerset of S.

47.4 Cardinality

Main article: Cardinality

The cardinality | S | of a set S is “the number of members of S.” For example, if B = blue, white, red, | B | = 3.There is a unique set with no members and zero cardinality, which is called the empty set (or the null set) and isdenoted by the symbol ∅ (other notations are used; see empty set). For example, the set of all three-sided squares haszero members and thus is the empty set. Though it may seem trivial, the empty set, like the number zero, is importantin mathematics; indeed, the existence of this set is one of the fundamental concepts of axiomatic set theory.Some sets have infinite cardinality. The set N of natural numbers, for instance, is infinite. Some infinite cardinalitiesare greater than others. For instance, the set of real numbers has greater cardinality than the set of natural numbers.However, it can be shown that the cardinality of (which is to say, the number of points on) a straight line is the sameas the cardinality of any segment of that line, of the entire plane, and indeed of any finite-dimensional Euclideanspace.

47.5 Special sets

There are some sets that hold great mathematical importance and are referred to with such regularity that they haveacquired special names and notational conventions to identify them. One of these is the empty set, denoted or ∅.Another is the unit set x, which contains exactly one element, namely x.[2] Many of these sets are represented usingblackboard bold or bold typeface. Special sets of numbers include

• P or ℙ, denoting the set of all primes: P = 2, 3, 5, 7, 11, 13, 17, ....

• N or ℕ, denoting the set of all natural numbers: N = 1, 2, 3, . . . (sometimes defined containing 0).

47.6. BASIC OPERATIONS 201

• Z or ℤ, denoting the set of all integers (whether positive, negative or zero): Z = ..., −2, −1, 0, 1, 2, ....

• Q or ℚ, denoting the set of all rational numbers (that is, the set of all proper and improper fractions): Q = a/b: a, b ∈ Z, b ≠ 0. For example, 1/4 ∈ Q and 11/6 ∈ Q. All integers are in this set since every integer a can beexpressed as the fraction a/1 (Z ⊊ Q).

• R or ℝ, denoting the set of all real numbers. This set includes all rational numbers, together with all irrationalnumbers (that is, numbers that cannot be rewritten as fractions, such as √2, as well as transcendental numberssuch as π, e and numbers that cannot be defined).

• C or ℂ, denoting the set of all complex numbers: C = a + bi : a, b ∈ R. For example, 1 + 2i ∈ C.

• H or ℍ, denoting the set of all quaternions: H = a + bi + cj + dk : a, b, c, d ∈ R. For example, 1 + i + 2j −k ∈ H.

Positive and negative sets are denoted by a superscript - or +. For example ℚ+ represents the set of positive rationalnumbers.Each of the above sets of numbers has an infinite number of elements, and each can be considered to be a propersubset of the sets listed below it. The primes are used less frequently than the others outside of number theory andrelated fields.

47.6 Basic operations

There are several fundamental operations for constructing new sets from given sets.

47.6.1 Unions

The union of A and B, denoted A ∪ B

202 CHAPTER 47. SET (MATHEMATICS)

Main article: Union (set theory)

Two sets can be “added” together. The union of A and B, denoted by A ∪ B, is the set of all things that are membersof either A or B.Examples:

• 1, 2 ∪ 1, 2 = 1, 2.• 1, 2 ∪ 2, 3 = 1, 2, 3.• 1, 2, 3 ∪ 3, 4, 5 = 1, 2, 3, 4, 5

Some basic properties of unions:

• A ∪ B = B ∪ A.• A ∪ (B ∪ C) = (A ∪ B) ∪ C.• A ⊆ (A ∪ B).• A ∪ A = A.• A ∪ ∅ = A.• A ⊆ B if and only if A ∪ B = B.

47.6.2 Intersections

Main article: Intersection (set theory)

A new set can also be constructed by determining which members two sets have “in common”. The intersection of Aand B, denoted by A ∩ B, is the set of all things that are members of both A and B. If A ∩ B = ∅, then A and B aresaid to be disjoint.Examples:

• 1, 2 ∩ 1, 2 = 1, 2.• 1, 2 ∩ 2, 3 = 2.

Some basic properties of intersections:

• A ∩ B = B ∩ A.• A ∩ (B ∩ C) = (A ∩ B) ∩ C.• A ∩ B ⊆ A.• A ∩ A = A.• A ∩ ∅ = ∅.• A ⊆ B if and only if A ∩ B = A.

47.6.3 Complements

Main article: Complement (set theory)

Two sets can also be “subtracted”. The relative complement of B in A (also called the set-theoretic difference of A andB), denoted by A \ B (or A − B), is the set of all elements that are members of A but not members of B. Note that itis valid to “subtract” members of a set that are not in the set, such as removing the element green from the set 1, 2,3; doing so has no effect.In certain settings all sets under discussion are considered to be subsets of a given universal set U. In such cases, U \A is called the absolute complement or simply complement of A, and is denoted by A′.Examples:

47.6. BASIC OPERATIONS 203

The intersection of A and B, denoted A ∩ B.

• 1, 2 \ 1, 2 = ∅.

• 1, 2, 3, 4 \ 1, 3 = 2, 4.

• If U is the set of integers, E is the set of even integers, and O is the set of odd integers, then U \ E= E′ = O.

Some basic properties of complements:

• A \ B ≠ B \ A for A ≠ B.

• A ∪ A′ = U.

• A ∩ A′ = ∅.

• (A′)′ = A.

• A \ A = ∅.

• U′ = ∅ and ∅′ = U.

• A \ B = A ∩ B′.

An extension of the complement is the symmetric difference, defined for sets A, B as

A∆B = (A \B) ∪ (B \A).

For example, the symmetric difference of 7,8,9,10 and 9,10,11,12 is the set 7,8,11,12.

204 CHAPTER 47. SET (MATHEMATICS)

The relative complementof B in A

47.6.4 Cartesian product

Main article: Cartesian product

A new set can be constructed by associating every element of one set with every element of another set. The Cartesianproduct of two sets A and B, denoted by A × B is the set of all ordered pairs (a, b) such that a is a member of A andb is a member of B.Examples:

• 1, 2 × red, white = (1, red), (1, white), (2, red), (2, white).• 1, 2 × red, white, green = (1, red), (1, white), (1, green), (2, red), (2, white), (2, green) .• 1, 2 × 1, 2 = (1, 1), (1, 2), (2, 1), (2, 2).

Some basic properties of cartesian products:

• A × ∅ = ∅.• A × (B ∪ C) = (A × B) ∪ (A × C).• (A ∪ B) × C = (A × C) ∪ (B × C).

Let A and B be finite sets. Then

• | A × B | = | B × A | = | A | × | B |.

For example,

• a,b,c×d,e,f=(a,d),(a,e),(a,f),(b,d),(b,e),(b,f),(c,d),(c,e),(c,f).

47.7. APPLICATIONS 205

The complement of A in U

47.7 Applications

Set theory is seen as the foundation from which virtually all of mathematics can be derived. For example, structuresin abstract algebra, such as groups, fields and rings, are sets closed under one or more operations.One of the main applications of naive set theory is constructing relations. A relation from a domain A to a codomainB is a subset of the Cartesian product A × B. Given this concept, we are quick to see that the set F of all ordered pairs(x, x2), where x is real, is quite familiar. It has a domain set R and a codomain set that is also R, because the set of allsquares is subset of the set of all reals. If placed in functional notation, this relation becomes f(x) = x2. The reasonthese two are equivalent is for any given value, y that the function is defined for, its corresponding ordered pair, (y,y2) is a member of the set F.

47.8 Axiomatic set theory

Main article: Axiomatic set theory

Although initially naive set theory, which defines a set merely as any well-defined collection, was well accepted, itsoon ran into several obstacles. It was found that this definition spawned several paradoxes, most notably:

• Russell’s paradox—It shows that the “set of all sets that do not contain themselves,” i.e. the “set” x : x is a setand x ∉ x does not exist.

• Cantor’s paradox—It shows that “the set of all sets” cannot exist.

The reason is that the phrase well-defined is not very well defined. It was important to free set theory of theseparadoxes because nearly all of mathematics was being redefined in terms of set theory. In an attempt to avoid theseparadoxes, set theory was axiomatized based on first-order logic, and thus axiomatic set theory was born.

206 CHAPTER 47. SET (MATHEMATICS)

The symmetric difference of A and B

For most purposes however, naive set theory is still useful.

47.9 Principle of inclusion and exclusion

Main article: Inclusion-exclusion principle

This principle gives us the cardinality of the union of sets.

|A1 ∪A2 ∪A3 ∪ . . . ∪An| =(|A1|+ |A2|+ |A3|+ . . . |An|)−(|A1 ∩A2|+ |A1 ∩A3|+ . . . |An−1 ∩An|)+. . .+

(−1)n−1

(|A1 ∩A2 ∩A3 ∩ . . . ∩An|)

47.10 De Morgan’s Law

De Morgan stated two laws about Sets.If A and B are any two Sets then,

• (A ∪ B)′ = A′ ∩ B′

The complement of A union B equals the complement of A intersected with the complement of B.

• (A ∩ B)′ = A′ ∪ B′

The complement of A intersected with B is equal to the complement of A union to the complement of B.

47.11. SEE ALSO 207

47.11 See also• Set notation

• Mathematical object

• Alternative set theory

• Axiomatic set theory

• Category of sets

• Class (set theory)

• Dense set

• Family of sets

• Fuzzy set

• Internal set

• Mereology

• Multiset

• Naive set theory

• Principia Mathematica

• Rough set

• Russell’s paradox

• Sequence (mathematics)

• Taxonomy

• Tuple

47.12 Notes[1] “EineMenge, ist die Zusammenfassung bestimmter, wohlunterschiedenerObjekte unserer Anschauung oder unseresDenkens

– welche Elemente der Menge genannt werden – zu einem Ganzen.”

[2] Stoll, Robert. Sets, Logic and Axiomatic Theories. W. H. Freeman and Company. p. 5.

47.13 References• Dauben, JosephW., Georg Cantor: His Mathematics and Philosophy of the Infinite, Boston: Harvard UniversityPress (1979) ISBN 978-0-691-02447-9.

• Halmos, Paul R., Naive Set Theory, Princeton, N.J.: Van Nostrand (1960) ISBN 0-387-90092-6.

• Stoll, Robert R., Set Theory and Logic, Mineola, N.Y.: Dover Publications (1979) ISBN 0-486-63829-4.

• Velleman, Daniel, How To Prove It: A Structured Approach, Cambridge University Press (2006) ISBN 978-0-521-67599-4

47.14 External links• C2 Wiki – Examples of set operations using English operators.

• Mathematical Sets: Elements, Intersections & Unions, Education Portal Academy

Chapter 48

Strictly singular operator

In functional analysis, a branch of mathematics, a strictly singular operator is a bounded linear operator L from aBanach space X to another Banach space Y, such that it is not an isomorphism, and fails to be an isomorphism on anyinfinite dimensional subspace of X. Any compact operator is strictly singular, but not vice versa.[1][2]

Every bounded linear map T : lp → lq , for 1 ≤ q, p <∞ , p = q , is strictly singular. Here, lp and lq are sequencespaces. Similarly, every bounded linear map T : c0 → lp and T : lp → c0 , for 1 ≤ p < ∞ , is strictly singular.Here c0 is the Banach space of sequences converging to zero. This is a corollary of Pitt’s theorem, which states thatsuch T, for q < p, are compact.

48.1 References[1] N.L. Carothers, A Short Course on Banach Space Theory, (2005) London Mathematical Society Student Texts 64, Cam-

bridge University Press.

[2] C. J. Read, Strictly singular operators and the invariant subspace problem, Studia Mathematica 132 (3) (1999), 203-226.fulltext

208

Chapter 49

Subset

“Superset” redirects here. For other uses, see Superset (disambiguation).In mathematics, especially in set theory, a set A is a subset of a set B, or equivalently B is a superset of A, if A is

AB

Euler diagram showingA is a proper subset of B and conversely B is a proper superset of A

209

210 CHAPTER 49. SUBSET

“contained” inside B, that is, all elements of A are also elements of B. A and B may coincide. The relationship of oneset being a subset of another is called inclusion or sometimes containment.The subset relation defines a partial order on sets.The algebra of subsets forms a Boolean algebra in which the subset relation is called inclusion.

49.1 Definitions

If A and B are sets and every element of A is also an element of B, then:

• A is a subset of (or is included in) B, denoted by A ⊆ B ,or equivalently

• B is a superset of (or includes) A, denoted by B ⊇ A.

If A is a subset of B, but A is not equal to B (i.e. there exists at least one element of B which is not an element of A),then

• A is also a proper (or strict) subset of B; this is written as A ⊊ B.

or equivalently

• B is a proper superset of A; this is written as B ⊋ A.

For any set S, the inclusion relation ⊆ is a partial order on the set P(S) of all subsets of S (the power set of S).When quantified, A ⊆ B is represented as: ∀xx∈A → x∈B.[1]

49.2 ⊂ and ⊃ symbols

Some authors use the symbols ⊂ and ⊃ to indicate subset and superset respectively; that is, with the same meaningand instead of the symbols, ⊆ and ⊇.[2] So for example, for these authors, it is true of every set A that A ⊂ A.Other authors prefer to use the symbols ⊂ and ⊃ to indicate proper subset and superset, respectively, instead of ⊊ and⊋.[3] This usage makes ⊆ and ⊂ analogous to the inequality symbols ≤ and <. For example, if x ≤ y then x may ormay not equal y, but if x < y, then x may not equal y, and is less than y. Similarly, using the convention that ⊂ isproper subset, if A ⊆ B, then A may or may not equal B, but if A ⊂ B, then A definitely does not equal B.

49.3 Examples

• The set A = 1, 2 is a proper subset of B = 1, 2, 3, thus both expressions A ⊆ B and A ⊊ B are true.

• The set D = 1, 2, 3 is a subset of E = 1, 2, 3, thus D ⊆ E is true, and D ⊊ E is not true (false).

• Any set is a subset of itself, but not a proper subset. (X ⊆ X is true, and X ⊊ X is false for any set X.)

• The empty set , denoted by ∅, is also a subset of any given set X. It is also always a proper subset of any setexcept itself.

• The set x: x is a prime number greater than 10 is a proper subset of x: x is an odd number greater than 10

• The set of natural numbers is a proper subset of the set of rational numbers; likewise, the set of points in a linesegment is a proper subset of the set of points in a line. These are two examples in which both the subset andthe whole set are infinite, and the subset has the same cardinality (the concept that corresponds to size, that is,the number of elements, of a finite set) as the whole; such cases can run counter to one’s initial intuition.

• The set of rational numbers is a proper subset of the set of real numbers. In this example, both sets are infinitebut the latter set has a larger cardinality (or power) than the former set.

49.4. OTHER PROPERTIES OF INCLUSION 211

polygonsregular

polygons

The regular polygons form a subset of the polygons

Another example in an Euler diagram:

• A is a proper subset of B

• C is a subset but no proper subset of B

49.4 Other properties of inclusion

Inclusion is the canonical partial order in the sense that every partially ordered set (X, ⪯ ) is isomorphic to somecollection of sets ordered by inclusion. The ordinal numbers are a simple example—if each ordinal n is identifiedwith the set [n] of all ordinals less than or equal to n, then a ≤ b if and only if [a] ⊆ [b].For the power set P(S) of a set S, the inclusion partial order is (up to an order isomorphism) the Cartesian product ofk = |S| (the cardinality of S) copies of the partial order on 0,1 for which 0 < 1. This can be illustrated by enumeratingS = s1, s2, …, sk and associating with each subset T ⊆ S (which is to say with each element of 2S) the k-tuple from0,1k of which the ith coordinate is 1 if and only if si is a member of T.

49.5 See also• Containment order

49.6 References[1] Rosen, Kenneth H. (2012). Discrete Mathematics and Its Applications (7th ed.). New York: McGraw-Hill. p. 119. ISBN

978-0-07-338309-5.

[2] Rudin, Walter (1987), Real and complex analysis (3rd ed.), New York: McGraw-Hill, p. 6, ISBN 978-0-07-054234-1,MR 924157

212 CHAPTER 49. SUBSET

CB

A

A B and B C imply A C

[3] Subsets and Proper Subsets (PDF), retrieved 2012-09-07

• Jech, Thomas (2002). Set Theory. Springer-Verlag. ISBN 3-540-44085-2.

49.7 External links• Weisstein, Eric W., “Subset”, MathWorld.

Chapter 50

Subspace topology

In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which isequipped with a topology induced from that of X called the subspace topology (or the relative topology, or theinduced topology, or the trace topology).

50.1 Definition

Given a topological space (X, τ) and a subset S of X , the subspace topology on S is defined by

τS = S ∩ U | U ∈ τ.

That is, a subset of S is open in the subspace topology if and only if it is the intersection of S with an open set in(X, τ) . If S is equipped with the subspace topology then it is a topological space in its own right, and is called asubspace of (X, τ) . Subsets of topological spaces are usually assumed to be equipped with the subspace topologyunless otherwise stated.Alternatively we can define the subspace topology for a subset S ofX as the coarsest topology for which the inclusionmap

ι : S → X

is continuous.More generally, suppose i is an injection from a set S to a topological space X . Then the subspace topology on Sis defined as the coarsest topology for which i is continuous. The open sets in this topology are precisely the ones ofthe form i−1(U) for U open inX . S is then homeomorphic to its image inX (also with the subspace topology) andi is called a topological embedding.

50.2 Examples

In the following, R represents the real numbers with their usual topology.

• The subspace topology of the natural numbers, as a subspace of R , is the discrete topology.

• The rational numbers Q considered as a subspace of R do not have the discrete topology (the point 0 forexample is not an open set in Q ). If a and b are rational, then the intervals (a, b) and [a, b] are respectivelyopen and closed, but if a and b are irrational, then the set of all x with a < x < b is both open and closed.

• The set [0,1] as a subspace of R is both open and closed, whereas as a subset of R it is only closed.

213

214 CHAPTER 50. SUBSPACE TOPOLOGY

• As a subspace of R , [0, 1] ∪ [2, 3] is composed of two disjoint open subsets (which happen also to be closed),and is therefore a disconnected space.

• Let S = [0, 1) be a subspace of the real line R . Then [0, 1/2) is open in S but not in R . Likewise [½, 1) isclosed in S but not in R . S is both open and closed as a subset of itself but not as a subset of R .

50.3 Properties

The subspace topology has the following characteristic property. Let Y be a subspace of X and let i : Y → X bethe inclusion map. Then for any topological space Z a map f : Z → Y is continuous if and only if the compositemap i f is continuous.

Characteristic property of the subspace topology

This property is characteristic in the sense that it can be used to define the subspace topology on Y .We list some further properties of the subspace topology. In the following let S be a subspace of X .

50.4. PRESERVATION OF TOPOLOGICAL PROPERTIES 215

• If f : X → Y is continuous the restriction to S is continuous.

• If f : X → Y is continuous then f : X → f(X) is continuous.

• The closed sets in S are precisely the intersections of S with closed sets in X .

• If A is a subspace of S then A is also a subspace of X with the same topology. In other words the subspacetopology that A inherits from S is the same as the one it inherits fromX .

• Suppose S is an open subspace of X . Then a subset of S is open in S if and only if it is open in X .

• Suppose S is a closed subspace of X . Then a subset of S is closed in S if and only if it is closed in X .

• If B is a basis for X then BS = U ∩ S : U ∈ B is a basis for S .

• The topology induced on a subset of a metric space by restricting the metric to this subset coincides withsubspace topology for this subset.

50.4 Preservation of topological properties

If a topological space having some topological property implies its subspaces have that property, then we say theproperty is hereditary. If only closed subspaces must share the property we call it weakly hereditary.

• Every open and every closed subspace of a completely metrizable space is completely metrizable.

• Every open subspace of a Baire space is a Baire space.

• Every closed subspace of a compact space is compact.

• Being a Hausdorff space is hereditary.

• Being a normal space is weakly hereditary.

• Total boundedness is hereditary.

• Being totally disconnected is hereditary.

• First countability and second countability are hereditary.

50.5 See also• the dual notion quotient space

• product topology

• direct sum topology

50.6 References• Bourbaki, Nicolas, Elements of Mathematics: General Topology, Addison-Wesley (1966)

• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 507446

• Willard, Stephen. General Topology, Dover Publications (2004) ISBN 0-486-43479-6

Chapter 51

Supercompact space

In mathematics, in the field of topology, a topological space is called supercompact if there is a subbasis such thatevery open cover of the topological space from elements of the subbasis has a subcover with at most two subbasiselements. Supercompactness and the related notion of superextension was introduced by J. de Groot in 1967.

51.1 Examples

By the Alexander subbase theorem, every supercompact space is compact. Conversely, many (but not all) compactspaces are supercompact. The following are examples of supercompact spaces:

• Compact linearly ordered spaces with the order topology and all continuous images of such spaces (Bula et al.1992)

• Compact metrizable spaces (due originally to M. Strok and A. Szymański 1975, see also Mills 1979)

• A product of supercompact spaces is supercompact (like a similar statement about compactness, Tychonoff’stheorem, it is equivalent to the axiom of choice, Banaschewski 1993)

51.2 Some Properties

Some compact Hausdorff spaces are not supercompact; such an example is given by the Stone–Čech compactificationof the natural numbers (with the discrete topology) (Bell 1978).A continuous image of a supercompact space need not be supercompact (Verbeek 1972, Mills—van Mill 1979).In a supercompact space (or any continuous image of one), the cluster point of any countable subset is the limit of anontrivial convergent sequence. (Yang 1994)

51.3 References

• B. Banaschewski, “Supercompactness, products and the axiom of choice.” Kyungpook Math. J. 33 (1993), no.1, 111—114.

• Bula, W.; Nikiel, J.; Tuncali, H. M.; Tymchatyn, E. D. “Continuous images of ordered compacta are regularsupercompact.” Proceedings of the Tsukuba Topology Symposium (Tsukuba, 1990). Topology Appl. 45(1992), no. 3, 203—221.

• Murray G. Bell. “Not all compact Hausdorff spaces are supercompact.” General Topology and Appl. 8 (1978),no. 2, 151—155.

216

51.3. REFERENCES 217

• J. de Groot, “Supercompactness and superextensions.” Contributions to extension theory of topological struc-tures. Proceedings of the Symposium held in Berlin, August 14—19, 1967. Edited by J. Flachsmeyer, H.Poppe and F. Terpe. VEB Deutscher Verlag der Wissenschaften, Berlin 1969 279 pp.

• Engelking, R (1977), General topology, Taylor & Francis, ISBN 978-0-8002-0209-5.

• Malykhin, VI; Ponomarev, VI (1977), “General topology (set-theoretic trend)", Journal of Mathematical Sci-ences (New York: Springer) 7 (4): 587–629, doi:10.1007/BF01084982

• Mills, Charles F. (1979), “A simpler proof that compact metric spaces are supercompact”, Proceedings of theAmerican Mathematical Society (Proceedings of the American Mathematical Society, Vol. 73, No. 3) 73 (3):388–390, doi:10.2307/2042369, JSTOR 2042369, MR 518526

• Mills, Charles F.; van Mill, Jan, “A nonsupercompact continuous image of a supercompact space.” Houston J.Math. 5 (1979), no. 2, 241—247.

• Mysior, Adam (1992), “Universal compact T1-spaces”, Canadian Mathematical Bulletin (Canadian Mathemat-ical Society) 35 (2): 261–266, doi:10.4153/CMB-1992-037-1.

• J. van Mill, Supercompactness and Wallman spaces. Mathematical Centre Tracts, No. 85. MathematischCentrum, Amsterdam, 1977. iv+238 pp. ISBN 90-6196-151-3

• M. Strok and A. Szymanski, "Compact metric spaces have binary bases. " Fund. Math. 89 (1975), no. 1,81—91.

• A. Verbeek, Superextensions of topological spaces. Mathematical Centre Tracts, No. 41. Mathematisch Cen-trum, Amsterdam, 1972. iv+155 pp.

• Yang, Zhong Qiang (1994), “All cluster points of countable sets in supercompact spaces are the limits ofnontrivial sequences”, Proceedings of the American Mathematical Society (Proceedings of the American Math-ematical Society, Vol. 122, No. 2) 122 (2): 591–595, doi:10.2307/2161053, JSTOR 2161053

Chapter 52

Topological space

In topology and related branches of mathematics, a topological space may be defined as a set of points, alongwith a set of neighbourhoods for each point, that satisfy a set of axioms relating points and neighbourhoods. Thedefinition of a topological space relies only upon set theory and is the most general notion of a mathematical spacethat allows for the definition of concepts such as continuity, connectedness, and convergence.[1] Other spaces, such asmanifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. Being sogeneral, topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics.The branch of mathematics that studies topological spaces in their own right is called point-set topology or generaltopology.

52.1 Definition

Main article: Characterizations of the category of topological spaces

The utility of the notion of a topology is shown by the fact that there are several equivalent definitions of this structure.Thus one chooses the axiomatisation suited for the application. The most commonly used, and the most elegant, isthat in terms of open sets, but the most intuitive is that in terms of neighbourhoods and so we give this first. Note: Avariety of more axiomatisations of topological spaces are listed in the Exercises of the book by Vaidyanathaswamy.

52.1.1 Neighbourhoods definition

This axiomatization is due to Felix Hausdorff. Let X be a set; the elements of X are usually called points, though theycan be any mathematical object. We allow X to be empty. Let N be a function assigning to each x (point) in X anon-empty collection N(x) of subsets of X. The elements of N(x) will be called neighbourhoods of x with respect toN (or, simply, neighbourhoods of x). The function N is called a neighbourhood topology if the axioms below[2] aresatisfied; and then X with N is called a topological space.

1. If N is a neighbourhood of x (i.e., N ∈ N(x)), then x ∈ N. In other words, each point belongs to every one of itsneighbourhoods.

2. If N is a subset of X and contains a neighbourhood of x, then N is a neighbourhood of x. I.e., every supersetof a neighbourhood of a point x in X is again a neighbourhood of x.

3. The intersection of two neighbourhoods of x is a neighbourhood of x.

4. Any neighbourhood N of x contains a neighbourhood M of x such that N is a neighbourhood of each point ofM.

The first three axioms for neighbourhoods have a clear meaning. The fourth axiom has a very important use in thestructure of the theory, that of linking together the neighbourhoods of different points of X.

218

52.1. DEFINITION 219

A standard example of such a system of neighbourhoods is for the real line R, where a subset N of R is defined to bea neighbourhood of a real number x if there is an open interval containing x and contained in N.Given such a structure, we can define a subset U of X to be open if U is a neighbourhood of all points in U. It is aremarkable fact that the open sets then satisfy the elegant axioms given below, and that, given these axioms, we canrecover the neighbourhoods satisfying the above axioms by defining N to be a neighbourhood of x if N contains anopen set U such that x ∈ U.[3]

52.1.2 Open sets definition

1 2 3 1 2 3

1 2 3 1 2 3

1 2 3 1 2 3

Four examples and two non-examples of topologies on the three-point set 1,2,3. The bottom-left example is not a topology becausethe union of 2 and 3 [i.e. 2,3] is missing; the bottom-right example is not a topology because the intersection of 1,2 and2,3 [i.e. 2], is missing.

A topological space is then a set X together with a collection of subsets of X, called open sets and satisfying thefollowing axioms:[4]

1. The empty set and X itself are open.

2. Any union of open sets is open.

3. The intersection of any finite number of open sets is open.

The collection τ of open sets is then also called a topology onX, or, if more precision is needed, an open set topology.The sets in τ are called the open sets, and their complements in X are called closed sets. A subset of Xmay be neitherclosed nor open, either closed or open, or both. A set that is both closed and open is called a clopen set.

220 CHAPTER 52. TOPOLOGICAL SPACE

Examples

1. X = 1, 2, 3, 4 and collection τ = , 1, 2, 3, 4 of only the two subsets of X required by the axioms forma topology, the trivial topology (indiscrete topology).

2. X = 1, 2, 3, 4 and collection τ = , 2, 1, 2, 2, 3, 1, 2, 3, 1, 2, 3, 4 of six subsets of X formanother topology.

3. X = 1, 2, 3, 4 and collection τ = P(X) (the power set of X) form a third topology, the discrete topology.

4. X =Z, the set of integers, and collection τ equal to all finite subsets of the integers plusZ itself is not a topology,because (for example) the union of all finite sets not containing zero is infinite but is not all of Z, and so is notin τ .

52.1.3 Closed sets definition

Using de Morgan’s laws, the above axioms defining open sets become axioms defining closed sets:

1. The empty set and X are closed.

2. The intersection of any collection of closed sets is also closed.

3. The union of any pair of closed sets is also closed.

Using these axioms, another way to define a topological space is as a set X together with a collection τ of closedsubsets of X. Thus the sets in the topology τ are the closed sets, and their complements in X are the open sets.

52.1.4 Other definitions

There are many other equivalent ways to define a topological space: in other words, the concepts of neighbourhoodor of open respectively closed set can be reconstructed from other starting points and satisfy the correct axioms.Another way to define a topological space is by using the Kuratowski closure axioms, which define the closed sets asthe fixed points of an operator on the power set of X.A net is a generalisation of the concept of sequence. A topology is completely determined if for every net in X theset of its accumulation points is specified.

52.2 Comparison of topologies

Main article: Comparison of topologies

A variety of topologies can be placed on a set to form a topological space. When every set in a topology τ1 is also ina topology τ2 and τ1 is a subset of τ2, we say that τ2 is finer than τ1, and τ1 is coarser than τ2. A proof that reliesonly on the existence of certain open sets will also hold for any finer topology, and similarly a proof that relies onlyon certain sets not being open applies to any coarser topology. The terms larger and smaller are sometimes used inplace of finer and coarser, respectively. The terms stronger and weaker are also used in the literature, but with littleagreement on the meaning, so one should always be sure of an author’s convention when reading.The collection of all topologies on a given fixed set X forms a complete lattice: if F = τα| α in A is a collectionof topologies on X, then the meet of F is the intersection of F, and the join of F is the meet of the collection of alltopologies on X that contain every member of F.

52.3 Continuous functions

A function f : X→ Y between topological spaces is called continuous if for all x ∈X and all neighbourhoodsN of f(x)there is a neighbourhoodM of x such that f(M) ⊆N. This relates easily to the usual definition in analysis. Equivalently,

52.4. EXAMPLES OF TOPOLOGICAL SPACES 221

f is continuous if the inverse image of every open set is open.[5] This is an attempt to capture the intuition that thereare no “jumps” or “separations” in the function. A homeomorphism is a bijection that is continuous and whose inverseis also continuous. Two spaces are called homeomorphic if there exists a homeomorphism between them. From thestandpoint of topology, homeomorphic spaces are essentially identical.In category theory, Top, the category of topological spaces with topological spaces as objects and continuous functionsas morphisms is one of the fundamental categories in category theory. The attempt to classify the objects of thiscategory (up to homeomorphism) by invariants has motivated areas of research, such as homotopy theory, homologytheory, and K-theory etc.

52.4 Examples of topological spaces

A given set may have many different topologies. If a set is given a different topology, it is viewed as a differenttopological space. Any set can be given the discrete topology in which every subset is open. The only convergentsequences or nets in this topology are those that are eventually constant. Also, any set can be given the trivial topology(also called the indiscrete topology), in which only the empty set and the whole space are open. Every sequence andnet in this topology converges to every point of the space. This example shows that in general topological spaces,limits of sequences need not be unique. However, often topological spaces must be Hausdorff spaces where limitpoints are unique.There are many ways of defining a topology on R, the set of real numbers. The standard topology on R is generatedby the open intervals. The set of all open intervals forms a base or basis for the topology, meaning that every openset is a union of some collection of sets from the base. In particular, this means that a set is open if there exists anopen interval of non zero radius about every point in the set. More generally, the Euclidean spaces Rn can be givena topology. In the usual topology on Rn the basic open sets are the open balls. Similarly, C, the set of complexnumbers, and Cn have a standard topology in which the basic open sets are open balls.Every metric space can be given a metric topology, in which the basic open sets are open balls defined by the metric.This is the standard topology on any normed vector space. On a finite-dimensional vector space this topology is thesame for all norms.Many sets of linear operators in functional analysis are endowed with topologies that are defined by specifying whena particular sequence of functions converges to the zero function.Any local field has a topology native to it, and this can be extended to vector spaces over that field.Every manifold has a natural topology since it is locally Euclidean. Similarly, every simplex and every simplicialcomplex inherits a natural topology from Rn.The Zariski topology is defined algebraically on the spectrum of a ring or an algebraic variety. On Rn or Cn, theclosed sets of the Zariski topology are the solution sets of systems of polynomial equations.A linear graph has a natural topology that generalises many of the geometric aspects of graphs with vertices andedges.The Sierpiński space is the simplest non-discrete topological space. It has important relations to the theory of com-putation and semantics.There exist numerous topologies on any given finite set. Such spaces are called finite topological spaces. Finite spacesare sometimes used to provide examples or counterexamples to conjectures about topological spaces in general.Any set can be given the cofinite topology in which the open sets are the empty set and the sets whose complementis finite. This is the smallest T1 topology on any infinite set.Any set can be given the cocountable topology, in which a set is defined as open if it is either empty or its complementis countable. When the set is uncountable, this topology serves as a counterexample in many situations.The real line can also be given the lower limit topology. Here, the basic open sets are the half open intervals [a, b).This topology on R is strictly finer than the Euclidean topology defined above; a sequence converges to a point in thistopology if and only if it converges from above in the Euclidean topology. This example shows that a set may havemany distinct topologies defined on it.If Γ is an ordinal number, then the set Γ = [0, Γ) may be endowed with the order topology generated by the intervals(a, b), [0, b) and (a, Γ) where a and b are elements of Γ.

222 CHAPTER 52. TOPOLOGICAL SPACE

52.5 Topological constructions

Every subset of a topological space can be given the subspace topology in which the open sets are the intersectionsof the open sets of the larger space with the subset. For any indexed family of topological spaces, the product can begiven the product topology, which is generated by the inverse images of open sets of the factors under the projectionmappings. For example, in finite products, a basis for the product topology consists of all products of open sets. Forinfinite products, there is the additional requirement that in a basic open set, all but finitely many of its projectionsare the entire space.A quotient space is defined as follows: if X is a topological space and Y is a set, and if f : X→ Y is a surjectivefunction, then the quotient topology on Y is the collection of subsets of Y that have open inverse images under f. Inother words, the quotient topology is the finest topology on Y for which f is continuous. A common example of aquotient topology is when an equivalence relation is defined on the topological space X. The map f is then the naturalprojection onto the set of equivalence classes.The Vietoris topology on the set of all non-empty subsets of a topological space X, named for Leopold Vietoris, isgenerated by the following basis: for every n-tuple U1, ..., Un of open sets in X, we construct a basis set consistingof all subsets of the union of the Ui that have non-empty intersections with each Ui.

52.6 Classification of topological spaces

Topological spaces can be broadly classified, up to homeomorphism, by their topological properties. A topologicalproperty is a property of spaces that is invariant under homeomorphisms. To prove that two spaces are not home-omorphic it is sufficient to find a topological property not shared by them. Examples of such properties includeconnectedness, compactness, and various separation axioms.See the article on topological properties for more details and examples.

52.7 Topological spaces with algebraic structure

For any algebraic objects we can introduce the discrete topology, under which the algebraic operations are continuousfunctions. For any such structure that is not finite, we often have a natural topology compatible with the algebraicoperations, in the sense that the algebraic operations are still continuous. This leads to concepts such as topologicalgroups, topological vector spaces, topological rings and local fields.

52.8 Topological spaces with order structure

• Spectral. A space is spectral if and only if it is the prime spectrum of a ring (Hochster theorem).

• Specialization preorder. In a space the specialization (or canonical) preorder is defined by x ≤ y if andonly if clx ⊆ cly.

52.9 Specializations and generalizations

The following spaces and algebras are either more specialized or more general than the topological spaces discussedabove.

• Proximity spaces provide a notion of closeness of two sets.

• Metric spaces embody a metric, a precise notion of distance between points.

• Uniform spaces axiomatize ordering the distance between distinct points.

• A topological space in which the points are functions is called a function space.

52.10. SEE ALSO 223

• Cauchy spaces axiomatize the ability to test whether a net is Cauchy. Cauchy spaces provide a general settingfor studying completions.

• Convergence spaces capture some of the features of convergence of filters.

• Grothendieck sites are categories with additional data axiomatizing whether a family of arrows covers an object.Sites are a general setting for defining sheaves.

52.10 See also

• Space (mathematics)

• Kolmogorov space (T0)

• accessible/Fréchet space (T1)

• Hausdorff space (T2)

• Completely Hausdorff space and Urysohn space (T₂½)

• Regular space and regular Hausdorff space (T3)

• Tychonoff space and completely regular space (T₃½)

• Normal Hausdorff space (T4)

• Completely normal Hausdorff space (T5)

• Perfectly normal Hausdorff space (T6)

• Quasitopological space

• Complete Heyting algebra – The system of all open sets of a given topological space ordered by inclusion is acomplete Heyting algebra.

52.11 Notes[1] Schubert 1968, p. 13

[2] Brown 2006, section 2.1.

[3] Brown 2006, section 2.2.

[4] Armstrong 1983, definition 2.1.

[5] Armstrong 1983, theorem 2.6.

52.12 References

• Armstrong, M. A. (1983) [1979]. Basic Topology. Undergraduate texts in mathematics. Springer. ISBN0-387-90839-0.

• Bredon, Glen E., Topology and Geometry (Graduate Texts in Mathematics), Springer; 1st edition (October 17,1997). ISBN 0-387-97926-3.

• Bourbaki, Nicolas; Elements of Mathematics: General Topology, Addison-Wesley (1966).

• Brown, Ronald, Topology and groupoids, Booksurge (2006) ISBN 1-4196-2722-8 (3rd edition of differentlytitled books) (order from amazon.com).

• Čech, Eduard; Point Sets, Academic Press (1969).

224 CHAPTER 52. TOPOLOGICAL SPACE

• Fulton, William, Algebraic Topology, (Graduate Texts in Mathematics), Springer; 1st edition (September 5,1997). ISBN 0-387-94327-7.

• Lipschutz, Seymour; Schaum’s Outline of General Topology, McGraw-Hill; 1st edition (June 1, 1968). ISBN0-07-037988-2.

• Munkres, James; Topology, Prentice Hall; 2nd edition (December 28, 1999). ISBN 0-13-181629-2.

• Runde, Volker; A Taste of Topology (Universitext), Springer; 1st edition (July 6, 2005). ISBN 0-387-25790-X.

• Schubert, Horst (1968), Topology, Allyn and Bacon

• Steen, Lynn A. and Seebach, J. Arthur Jr.; Counterexamples in Topology, Holt, Rinehart and Winston (1970).ISBN 0-03-079485-4.

• Vaidyanathaswamy, R. (1960). Set Topology. Chelsea Publishing Co. ISBN 0486404560.

• Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.

52.13 External links• Hazewinkel, Michiel, ed. (2001), “Topological space”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Topological space at PlanetMath.org.

Chapter 53

Topology

Not to be confused with topography.This article is about the branch of mathematics. For other uses, see Topology (disambiguation).In mathematics, topology (from the Greek τόπος, “place”, and λόγος, “study”), the study of topological spaces, is

Möbius strips, which have only one surface and one edge, are a kind of object studied in topology.

an area of mathematics concerned with the properties of space that are preserved under continuous deformations,such as stretching and bending, but not tearing or gluing. Important topological properties include connectedness andcompactness.Topology developed as a field of study out of geometry and set theory, through analysis of such concepts as space,dimension, and transformation. Such ideas go back to Gottfried Leibniz, who in the 17th century envisioned thegeometria situs (Greek-Latin for “geometry of place”) and analysis situs (Greek-Latin for “picking apart of place”).The term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the firstdecades of the 20th century that the idea of a topological space was developed. By the middle of the 20th century,topology had become a major branch of mathematics.Topology has many subfields:

• General topology establishes the foundational aspects of topology and investigates properties of topological

225

226 CHAPTER 53. TOPOLOGY

spaces and investigates concepts inherent to topological spaces. It includes point-set topology, which is thefoundational topology used in all other branches (including topics like compactness and connectedness).

• Algebraic topology tries to measure degrees of connectivity using algebraic constructs such as homology andhomotopy groups.

• Differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closelyrelated to differential geometry and together they make up the geometric theory of differentiable manifolds.

• Geometric topology primarily studies manifolds and their embeddings (placements) in other manifolds. Aparticularly active area is low dimensional topology, which studies manifolds of four or fewer dimensions.This includes knot theory, the study of mathematical knots.

A three-dimensional depiction of a thickened trefoil knot, the simplest non-trivial knot

See also: topology glossary for definitions of some of the terms used in topology, and topological space for a moretechnical treatment of the subject.

53.1 History

Topology began with the investigation of certain questions in geometry. Leonhard Euler's 1736 paper on the SevenBridges of Königsberg[1] is regarded as one of the first academic treatises in modern topology.The term “Topologie” was introduced in German in 1847 by Johann Benedict Listing in Vorstudien zur Topologie,[2]who had used the word for ten years in correspondence before its first appearance in print. The English form topologywas first used in 1883 in Listing’s obituary in the journal Nature[3] to distinguish “qualitative geometry from theordinary geometry in which quantitative relations chiefly are treated”. The term topologist in the sense of a specialistin topology was used in 1905 in the magazine Spectator. However, none of these uses corresponds exactly to themodern definition of topology.

53.2. INTRODUCTION 227

The Seven Bridges of Königsberg was a problem solved by Euler.

Modern topology depends strongly on the ideas of set theory, developed by Georg Cantor in the later part of the 19thcentury. In addition to establishing the basic ideas of set theory, Cantor considered point sets in Euclidean space aspart of his study of Fourier series.Henri Poincaré published Analysis Situs in 1895,[4] introducing the concepts of homotopy and homology, which arenow considered part of algebraic topology.Unifying the work on function spaces of Georg Cantor, Vito Volterra, Cesare Arzelà, Jacques Hadamard, GiulioAscoli and others, Maurice Fréchet introduced the metric space in 1906.[5] A metric space is now considered aspecial case of a general topological space. In 1914, Felix Hausdorff coined the term “topological space” and gavethe definition for what is now called a Hausdorff space.[6] Currently, a topological space is a slight generalization ofHausdorff spaces, given in 1922 by Kazimierz Kuratowski.For further developments, see point-set topology and algebraic topology.

53.2 Introduction

Topology can be formally defined as “the study of qualitative properties of certain objects (called topological spaces)that are invariant under a certain kind of transformation (called a continuous map), especially those properties thatare invariant under a certain kind of transformation (called homeomorphism).”Topology is also used to refer to a structure imposed upon a set X, a structure that essentially 'characterizes’ the set Xas a topological space by taking proper care of properties such as convergence, connectedness and continuity, upontransformation.Topological spaces show up naturally in almost every branch of mathematics. This has made topology one of thegreat unifying ideas of mathematics.

228 CHAPTER 53. TOPOLOGY

The motivating insight behind topology is that some geometric problems depend not on the exact shape of the objectsinvolved, but rather on the way they are put together. For example, the square and the circle have many properties incommon: they are both one dimensional objects (from a topological point of view) and both separate the plane intotwo parts, the part inside and the part outside.One of the first papers in topology was the demonstration, by Leonhard Euler, that it was impossible to find a routethrough the town of Königsberg (now Kaliningrad) that would cross each of its seven bridges exactly once. Thisresult did not depend on the lengths of the bridges, nor on their distance from one another, but only on connectivityproperties: which bridges are connected to which islands or riverbanks. This problem in introductory mathematicscalled Seven Bridges of Königsberg led to the branch of mathematics known as graph theory.

A continuous deformation (a type of homeomorphism) of a mug into a doughnut (torus) and back

Similarly, the hairy ball theorem of algebraic topology says that “one cannot comb the hair flat on a hairy ball withoutcreating a cowlick.” This fact is immediately convincing to most people, even though they might not recognize themore formal statement of the theorem, that there is no nonvanishing continuous tangent vector field on the sphere.As with the Bridges of Königsberg, the result does not depend on the shape of the sphere; it applies to any kind ofsmooth blob, as long as it has no holes.To deal with these problems that do not rely on the exact shape of the objects, one must be clear about just whatproperties these problems do rely on. From this need arises the notion of homeomorphism. The impossibility ofcrossing each bridge just once applies to any arrangement of bridges homeomorphic to those in Königsberg, and thehairy ball theorem applies to any space homeomorphic to a sphere.

53.3. CONCEPTS 229

Intuitively, two spaces are homeomorphic if one can be deformed into the other without cutting or gluing. A traditionaljoke is that a topologist cannot distinguish a coffee mug from a doughnut, since a sufficiently pliable doughnut couldbe reshaped to a coffee cup by creating a dimple and progressively enlarging it, while shrinking the hole into a handle.Homeomorphism can be considered the most basic topological equivalence. Another is homotopy equivalence. Thisis harder to describe without getting technical, but the essential notion is that two objects are homotopy equivalent ifthey both result from “squishing” some larger object.An introductory exercise is to classify the uppercase letters of the English alphabet according to homeomorphismand homotopy equivalence. The result depends partially on the font used. The figures use the sans-serif Myriad font.Homotopy equivalence is a rougher relationship than homeomorphism; a homotopy equivalence class can containseveral homeomorphism classes. The simple case of homotopy equivalence described above can be used here toshow two letters are homotopy equivalent. For example, O fits inside P and the tail of the P can be squished to the“hole” part.Homeomorphism classes are:

• no holes,

• no holes three tails,

• no holes four tails,

• one hole no tail,

• one hole one tail,

• one hole two tails,

• two holes no tail, and

• a bar with four tails (the “bar” on the K is almost too short to see).

Homotopy classes are larger, because the tails can be squished down to a point. They are:

• one hole,

• two holes, and

• no holes.

To be sure that the letters are classified correctly, we need to show that two letters in the same class are equivalent andtwo letters in different classes are not equivalent. In the case of homeomorphism, this can be done by selecting pointsand showing their removal disconnects the letters differently. For example, X and Y are not homeomorphic becauseremoving the center point of the X leaves four pieces; whatever point in Y corresponds to this point, its removalcan leave at most three pieces. The case of homotopy equivalence is harder and requires a more elaborate argumentshowing an algebraic invariant, such as the fundamental group, is different on the supposedly differing classes.Letter topology has practical relevance in stencil typography. For instance, Braggadocio font stencils are made of oneconnected piece of material.

53.3 Concepts

53.3.1 Topologies on Sets

Main article: Topological space

The term topology also refers to a specific mathematical idea which is central to the area of mathematics calledtopology. Informally, a topology is used to tell how elements of a set are related spatially to each other. The same setcan have different topologies. For instance, the real line, the complex plane, and the Cantor set can be thought of asthe same set with different topologies.Formally, let X be a set and let τ be a family of subsets of X. Then τ is called a topology on X if:

230 CHAPTER 53. TOPOLOGY

1. Both the empty set and X are elements of τ

2. Any union of elements of τ is an element of τ

3. Any intersection of finitely many elements of τ is an element of τ

If τ is a topology on X, then the pair (X, τ) is called a topological space. The notation Xτmay be used to denote a setX endowed with the particular topology τ.The members of τ are called open sets in X. A subset of X is said to be closed if its complement is in τ (i.e., itscomplement is open). A subset of X may be open, closed, both (clopen set), or neither. The empty set and X itselfare always both closed and open. An open set containing a point x is called a 'neighborhood' of x.A set with a topology is called a topological space.

53.3.2 Continuous functions and homeomorphisms

Main articles: Continuous function and homeomorphism

A function or map from one topological space to another is called continuous if the inverse image of any open set isopen. If the function maps the real numbers to the real numbers (both spaces with the Standard Topology), then thisdefinition of continuous is equivalent to the definition of continuous in calculus. If a continuous function is one-to-oneand onto, and if the inverse of the function is also continuous, then the function is called a homeomorphism and thedomain of the function is said to be homeomorphic to the range. Another way of saying this is that the function hasa natural extension to the topology. If two spaces are homeomorphic, they have identical topological properties, andare considered topologically the same. The cube and the sphere are homeomorphic, as are the coffee cup and thedoughnut. But the circle is not homeomorphic to the doughnut.

53.3.3 Manifolds

Main article: Manifold

While topological spaces can be extremely varied and exotic, many areas of topology focus on the more familiarclass of spaces known as manifolds. A manifold is a topological space that resembles Euclidean space near eachpoint. More precisely, each point of an n-dimensional manifold has a neighbourhood that is homeomorphic to theEuclidean space of dimension n. Lines and circles, but not figure eights, are one-dimensional manifolds. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, which can allbe realized in three dimensions, but also the Klein bottle and real projective plane which cannot.

53.4 Topics

53.4.1 General topology

Main article: General topology

General topology is the branch of topology dealing with the basic set-theoretic definitions and constructions usedin topology.[7][8] It is the foundation of most other branches of topology, including differential topology, geometrictopology, and algebraic topology. Another name for general topology is point-set topology.The fundamental concepts in point-set topology are continuity, compactness, and connectedness. Intuitively, continu-ous functions take nearby points to nearby points; compact sets are those which can be covered by finitely many setsof arbitrarily small size; and connected sets are sets which cannot be divided into two pieces which are far apart. Thewords 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using open sets, as described below. Ifwe change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are.Each choice of definition for 'open set' is called a topology. A set with a topology is called a topological space.

53.4. TOPICS 231

Metric spaces are an important class of topological spaces where distances can be assigned a number called a metric.Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces.

53.4.2 Algebraic topology

Main article: Algebraic topology

Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces.[9]The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usuallymost classify up to homotopy equivalence.The most important of these invariants are homotopy groups, homology, and cohomology.Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraicproblems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroupof a free group is again a free group.

53.4.3 Differential topology

Main article: Differential topology

Differential topology is the field dealing with differentiable functions on differentiable manifolds.[10] It is closelyrelated to differential geometry and together they make up the geometric theory of differentiable manifolds.More specifically, differential topology considers the properties and structures that require only a smooth structureon a manifold to be defined. Smooth manifolds are 'softer' than manifolds with extra geometric structures, whichcan act as obstructions to certain types of equivalences and deformations that exist in differential topology. Forinstance, volume and Riemannian curvature are invariants that can distinguish different geometric structures on thesame smooth manifold—that is, one can smoothly “flatten out” certain manifolds, but it might require distorting thespace and affecting the curvature or volume.

53.4.4 Geometric topology

Main article: Geometric topology

Geometric topology is a branch of topology that primarily focuses on low-dimensional manifolds (i.e. dimensions2,3 and 4) and their interaction with geometry, but it also includes some higher-dimensional topology.[11] [12] Someexamples of topics in geometric topology are orientability, handle decompositions, local flatness, and the planar andhigher-dimensional Schönflies theorem.In high-dimensional topology, characteristic classes are a basic invariant, and surgery theory is a key theory.Low-dimensional topology is strongly geometric, as reflected in the uniformization theorem in 2 dimensions – ev-ery surface admits a constant curvature metric; geometrically, it has one of 3 possible geometries: positive curva-ture/spherical, zero curvature/flat, negative curvature/hyperbolic – and the geometrization conjecture (now theorem)in 3 dimensions – every 3-manifold can be cut into pieces, each of which has one of 8 possible geometries.2-dimensional topology can be studied as complex geometry in one variable (Riemann surfaces are complex curves)– by the uniformization theorem every conformal class of metrics is equivalent to a unique complex one, and 4-dimensional topology can be studied from the point of view of complex geometry in two variables (complex surfaces),though not every 4-manifold admits a complex structure.

53.4.5 Generalizations

Occasionally, one needs to use the tools of topology but a “set of points” is not available. In pointless topology oneconsiders instead the lattice of open sets as the basic notion of the theory,[13] while Grothendieck topologies are

232 CHAPTER 53. TOPOLOGY

structures defined on arbitrary categories that allow the definition of sheaves on those categories, and with that thedefinition of general cohomology theories.[14]

53.5 Applications

53.5.1 Biology

Knot theory, a branch of topology, is used in biology to study the effects of certain enzymes on DNA. These enzymescut, twist, and reconnect the DNA, causing knotting with observable effects such as slower electrophoresis.[15] Topol-ogy is also used in evolutionary biology to represent the relationship between phenotype and genotype.[16] Phenotypicforms which appear quite different can be separated by only a few mutations depending on how genetic changes mapto phenotypic changes during development.

53.5.2 Computer science

Topological data analysis uses techniques from algebraic topology to determine the large scale structure of a set (forinstance, determining if a cloud of points is spherical or toroidal). The main method used by topological data analysisis:

1. Replace a set of data points with a family of simplicial complexes, indexed by a proximity parameter.

2. Analyse these topological complexes via algebraic topology — specifically, via the theory of persistent homol-ogy.[17]

3. Encode the persistent homology of a data set in the form of a parameterized version of a Betti number whichis called a barcode.[17]

53.5.3 Physics

In physics, topology is used in several areas such as quantum field theory and cosmology.A topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which com-putes topological invariants.Although TQFTs were invented by physicists, they are also of mathematical interest, being related to, among otherthings, knot theory and the theory of four-manifolds in algebraic topology, and to the theory of moduli spaces inalgebraic geometry. Donaldson, Jones, Witten, and Kontsevich have all won Fields Medals for work related to topo-logical field theory.In cosmology, topology can be used to describe the overall shape of the universe.[18] This area is known as spacetimetopology.

53.5.4 Robotics

The various possible positions of a robot can be described by a manifold called configuration space.[19] In the area ofmotion planning, one finds paths between two points in configuration space. These paths represent a motion of therobot’s joints and other parts into the desired location and pose.

53.6 See also

• Equivariant topology

• General topology

• List of algebraic topology topics

53.7. REFERENCES 233

• List of examples in general topology

• List of general topology topics

• List of geometric topology topics

• List of topology topics

• Publications in topology

• Topology glossary

53.7 References

[1] Euler, Leonhard, Solutio problematis ad geometriam situs pertinentis

[2] Listing, Johann Benedict, “Vorstudien zur Topologie”, Vandenhoeck und Ruprecht, Göttingen, p. 67, 1848

[3] Tait, Peter Guthrie, “Johann Benedict Listing (obituary)", Nature *27*, 1 February 1883, pp. 316–317

[4] Poincaré, Henri, “Analysis situs”, Journal de l'École Polytechnique ser 2, 1 (1895) pp. 1–123

[5] Fréchet, Maurice, “Sur quelques points du calcul fonctionnel”, PhD dissertation, 1906

[6] Hausdorff, Felix, “Grundzüge der Mengenlehre”, Leipzig: Veit. In (Hausdorff Werke, II (2002), 91–576)

[7] Munkres, James R. Topology. Vol. 2. Upper Saddle River: Prentice Hall, 2000.

[8] Adams, Colin Conrad, and Robert David Franzosa. Introduction to topology: pure and applied. Pearson Prentice Hall,2008.

[9] Allen Hatcher, Algebraic topology. (2002) Cambridge University Press, xii+544 pp. ISBN 0-521-79160-X and ISBN0-521-79540-0.

[10] Lee, John M. (2006). Introduction to Smooth Manifolds. Springer-Verlag. ISBN 978-0-387-95448-6.

[11] Budney, Ryan (2011). “What is geometric topology?". mathoverflow.net. Retrieved 29 December 2013.

[12] R.B. Sher and R.J. Daverman (2002), Handbook of Geometric Topology, North-Holland. ISBN 0-444-82432-4

[13] Johnstone, Peter T., 1983, "The point of pointless topology," Bulletin of the American Mathematical Society 8(1): 41-53.

[14] Artin, Michael (1962). Grothendieck topologies. Cambridge, MA: Harvard University, Dept. of Mathematics. Zbl0208.48701.

[15] Adams, Colin (2004). The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. American Math-ematical Society. ISBN 0-8218-3678-1

[16] Barble M R Stadler et al. “The Topology of the Possible: Formal Spaces Underlying Patterns of Evolutionary Change”.Journal of Theoretical Biology 213: 241–274. doi:10.1006/jtbi.2001.2423.

[17] Gunnar Carlsson (April 2009). “Topology and data” (PDF). BULLETIN (New Series) OF THE AMERICAN MATHEMAT-ICAL SOCIETY 46 (2): 255–308. doi:10.1090/S0273-0979-09-01249-X.

[18] The Shape of Space: How to Visualize Surfaces and Three-dimensional Manifolds 2nd ed (Marcel Dekker, 1985, ISBN0-8247-7437-X)

[19] John J. Craig, Introduction to Robotics: Mechanics and Control, 3rd Ed. Prentice-Hall, 2004

234 CHAPTER 53. TOPOLOGY

53.8 Further reading• Ryszard Engelking, General Topology, Heldermann Verlag, Sigma Series in Pure Mathematics, December1989, ISBN 3-88538-006-4.

• Bourbaki; Elements of Mathematics: General Topology, Addison–Wesley (1966).

• Breitenberger, E. (2006). “Johann Benedict Listing”. In James, I. M. History of Topology. North Holland.ISBN 978-0-444-82375-5.

• Kelley, John L. (1975). General Topology. Springer-Verlag. ISBN 0-387-90125-6.

• Brown, Ronald (2006). Topology and Groupoids. Booksurge. ISBN 1-4196-2722-8. (Provides a well mo-tivated, geometric account of general topology, and shows the use of groupoids in discussing van Kampen’stheorem, covering spaces, and orbit spaces.)

• Wacław Sierpiński, General Topology, Dover Publications, 2000, ISBN 0-486-41148-6

• Pickover, Clifford A. (2006). The Möbius Strip: Dr. August Möbius’s Marvelous Band in Mathematics, Games,Literature, Art, Technology, and Cosmology. Thunder’s Mouth Press. ISBN 1-56025-826-8. (Provides apopular introduction to topology and geometry)

• Gemignani, Michael C. (1990) [1967], Elementary Topology (2nd ed.), Dover Publications Inc., ISBN 0-486-66522-4

53.9 External links• Hazewinkel, Michiel, ed. (2001), “Topology, general”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Elementary Topology: A First Course Viro, Ivanov, Netsvetaev, Kharlamov.

• Topology at DMOZ

• The Topological Zoo at The Geometry Center.

• Topology Atlas

• Topology Course Lecture Notes Aisling McCluskey and Brian McMaster, Topology Atlas.

• Topology Glossary

• Moscow 1935: Topology moving towards America, a historical essay by Hassler Whitney.

Chapter 54

Total order

Inmathematics, a linear order, total order, simple order, or (non-strict) ordering is a binary relation (here denotedby infix ≤) on some setX which is transitive, antisymmetric, and total. A set paired with a total order is called a totallyordered set, a linearly ordered set, a simply ordered set, or a chain.If X is totally ordered under ≤, then the following statements hold for all a, b and c in X:

If a ≤ b and b ≤ a then a = b (antisymmetry);If a ≤ b and b ≤ c then a ≤ c (transitivity);a ≤ b or b ≤ a (totality).

Antisymmetry eliminates uncertain cases when both a precedes b and b precedes a.[1] A relation having the property of“totality” means that any pair of elements in the set of the relation are comparable under the relation. This also meansthat the set can be diagrammed as a line of elements, giving it the name linear.[2] Totality also implies reflexivity, i.e.,a ≤ a. Therefore, a total order is also a partial order. The partial order has a weaker form of the third condition. (Itrequires only reflexivity, not totality.) An extension of a given partial order to a total order is called a linear extensionof that partial order.

54.1 Strict total order

For each (non-strict) total order ≤ there is an associated asymmetric (hence irreflexive) relation <, called a strict totalorder, which can equivalently be defined in two ways:

• a < b if and only if a ≤ b and a ≠ b• a < b if and only if not b ≤ a (i.e., < is the inverse of the complement of ≤)

Properties:

• The relation is transitive: a < b and b < c implies a < c.• The relation is trichotomous: exactly one of a < b, b < a and a = b is true.• The relation is a strict weak order, where the associated equivalence is equality.

We can work the other way and start by choosing < as a transitive trichotomous binary relation; then a total order ≤can equivalently be defined in two ways:

• a ≤ b if and only if a < b or a = b• a ≤ b if and only if not b < a

Two more associated orders are the complements ≥ and >, completing the quadruple <, >, ≤, ≥.We can define or explain the way a set is totally ordered by any of these four relations; the notation implies whetherwe are talking about the non-strict or the strict total order.

235

236 CHAPTER 54. TOTAL ORDER

54.2 Examples

• The letters of the alphabet ordered by the standard dictionary order, e.g., A < B < C etc.

• Any subset of a totally ordered set, with the restriction of the order on the whole set.

• Any set of cardinal numbers or ordinal numbers (more strongly, these are well-orders).

• If X is any set and f an injective function from X to a totally ordered set then f induces a total ordering on Xby setting x1 < x2 if and only if f(x1) < f(x2).

• The lexicographical order on the Cartesian product of a set of totally ordered sets indexed by an ordinal, isitself a total order. For example, any set of words ordered alphabetically is a totally ordered set, viewed as asubset of a Cartesian product of a countable number of copies of a set formed by adding the space symbol tothe alphabet (and defining a space to be less than any letter).

• The set of real numbers ordered by the usual less than (<) or greater than (>) relations is totally ordered, hencealso the subsets of natural numbers, integers, and rational numbers. Each of these can be shown to be the unique(to within isomorphism) smallest example of a totally ordered set with a certain property, (a total order A isthe smallest with a certain property if whenever B has the property, there is an order isomorphism from A to asubset of B):

• The natural numbers comprise the smallest totally ordered set with no upper bound.• The integers comprise the smallest totally ordered set with neither an upper nor a lower bound.• The rational numbers comprise the smallest totally ordered set which is dense in the real numbers. Thedefinition of density used here says that for every 'a' and 'b' in the real numbers such that 'a' < 'b', there isa 'q' in the rational numbers such that 'a' < 'q' < 'b'.

• The real numbers comprise the smallest unbounded totally ordered set that is connected in the ordertopology (defined below).

• Ordered fields are totally ordered by definition. They include the rational numbers and the real numbers.

54.3 Further concepts

54.3.1 Chains

While chain is sometimes merely a synonym for totally ordered set, it can also refer to a totally ordered subset ofsome partially ordered set. The latter definition has a crucial role in Zorn’s lemma.For example, consider the set of all subsets of the integers partially ordered by inclusion. Then the set In : n is anatural number, where In is the set of natural numbers below n, is a chain in this ordering, as it is totally orderedunder inclusion: If n≤k, then In is a subset of Ik.

54.3.2 Lattice theory

One may define a totally ordered set as a particular kind of lattice, namely one in which we have

a ∨ b, a ∧ b = a, b for all a, b.

We then write a ≤ b if and only if a = a ∧ b . Hence a totally ordered set is a distributive lattice.

54.3. FURTHER CONCEPTS 237

54.3.3 Finite total orders

A simple counting argument will verify that any non-empty finite totally ordered set (and hence any non-empty subsetthereof) has a least element. Thus every finite total order is in fact a well order. Either by direct proof or by observingthat every well order is order isomorphic to an ordinal one may show that every finite total order is order isomorphicto an initial segment of the natural numbers ordered by <. In other words a total order on a set with k elements inducesa bijection with the first k natural numbers. Hence it is common to index finite total orders or well orders with ordertype ω by natural numbers in a fashion which respects the ordering (either starting with zero or with one).

54.3.4 Category theory

Totally ordered sets form a full subcategory of the category of partially ordered sets, with the morphisms being mapswhich respect the orders, i.e. maps f such that if a ≤ b then f(a) ≤ f(b).A bijective map between two totally ordered sets that respects the two orders is an isomorphism in this category.

54.3.5 Order topology

For any totally ordered set X we can define the open intervals (a, b) = x : a < x and x < b, (−∞, b) = x : x < b,(a, ∞) = x : a < x and (−∞, ∞) = X. We can use these open intervals to define a topology on any ordered set, theorder topology.When more than one order is being used on a set one talks about the order topology induced by a particular order.For instance if N is the natural numbers, < is less than and > greater than we might refer to the order topology onN induced by < and the order topology on N induced by > (in this case they happen to be identical but will not ingeneral).The order topology induced by a total order may be shown to be hereditarily normal.

54.3.6 Completeness

A totally ordered set is said to be complete if every nonempty subset that has an upper bound, has a least upperbound. For example, the set of real numbers R is complete but the set of rational numbers Q is not.There are a number of results relating properties of the order topology to the completeness of X:

• If the order topology on X is connected, X is complete.

• X is connected under the order topology if and only if it is complete and there is no gap in X (a gap is twopoints a and b in X with a < b such that no c satisfies a < c < b.)

• X is complete if and only if every bounded set that is closed in the order topology is compact.

A totally ordered set (with its order topology) which is a complete lattice is compact. Examples are the closed intervalsof real numbers, e.g. the unit interval [0,1], and the affinely extended real number system (extended real number line).There are order-preserving homeomorphisms between these examples.

54.3.7 Sums of orders

For any two disjoint total orders (A1,≤1) and (A2,≤2) , there is a natural order ≤+ on the set A1 ∪ A2 , which iscalled the sum of the two orders or sometimes just A1 +A2 :

For x, y ∈ A1 ∪A2 , x ≤+ y holds if and only if one of the following holds:

1. x, y ∈ A1 and x ≤1 y

2. x, y ∈ A2 and x ≤2 y

3. x ∈ A1 and y ∈ A2

238 CHAPTER 54. TOTAL ORDER

Intutitively, this means that the elements of the second set are added on top of the elements of the first set.More generally, if (I,≤) is a totally ordered index set, and for each i ∈ I the structure (Ai,≤i) is a linear order,where the sets Ai are pairwise disjoint, then the natural total order on

∪iAi is defined by

For x, y ∈∪

i∈I Ai , x ≤ y holds if:1. Either there is some i ∈ I with x ≤i y

2. or there are some i < j in I with x ∈ Ai , y ∈ Aj

54.4 Orders on the Cartesian product of totally ordered sets

In order of increasing strength, i.e., decreasing sets of pairs, three of the possible orders on the Cartesian product oftwo totally ordered sets are:

• Lexicographical order: (a,b) ≤ (c,d) if and only if a < c or (a = c and b ≤ d). This is a total order.• (a,b) ≤ (c,d) if and only if a ≤ c and b ≤ d (the product order). This is a partial order.• (a,b) ≤ (c,d) if and only if (a < c and b < d) or (a = c and b = d) (the reflexive closure of the direct product ofthe corresponding strict total orders). This is also a partial order.

All three can similarly be defined for the Cartesian product of more than two sets.Applied to the vector space Rn, each of these make it an ordered vector space.See also examples of partially ordered sets.A real function of n real variables defined on a subset of Rn defines a strict weak order and a corresponding totalpreorder on that subset.

54.5 Related structures

A binary relation that is antisymmetric, transitive, and reflexive (but not necessarily total) is a partial order.A group with a compatible total order is a totally ordered group.There are only a few nontrivial structures that are (interdefinable as) reducts of a total order. Forgetting the orientationresults in a betweenness relation. Forgetting the location of the ends results in a cyclic order. Forgetting both dataresults in a separation relation.[3]

54.6 See also• Order theory• Well-order• Suslin’s problem• Countryman line

54.7 Notes[1] Nederpelt, Rob (2004). “Chapter 20.2: Ordered Sets. Orderings”. Logical Reasoning: A First Course. Texts in Computing

3 (3rd, Revised ed.). King’s College Publications. p. 325. ISBN 0-9543006-7-X.

[2] Nederpelt, Rob (2004). “Chapter 20.3: Ordered Sets. Linear orderings”. Logical Reasoning: A First Course. Texts inComputing 3 (3rd, Revisied ed.). King’s College Publications. p. 330. ISBN 0-9543006-7-X.

[3] Macpherson, H. Dugald (2011), “A survey of homogeneous structures” (PDF),DiscreteMathematics, doi:10.1016/j.disc.2011.01.024,retrieved 28 April 2011

54.8. REFERENCES 239

54.8 References• George Grätzer (1971). Lattice theory: first concepts and distributive lattices. W. H. Freeman and Co. ISBN0-7167-0442-0

• John G. Hocking and Gail S. Young (1961). Topology. Corrected reprint, Dover, 1988. ISBN 0-486-65676-4

Chapter 55

Totally bounded space

In topology and related branches of mathematics, a totally bounded space is a space that can be covered by finitelymany subsets of any fixed “size” (where the meaning of “size” depends on the given context). The smaller the sizefixed, the more subsets may be needed, but any specific size should require only finitely many subsets. A relatednotion is a totally bounded set, in which only a subset of the space needs to be covered. Every subset of a totallybounded space is a totally bounded set; but even if a space is not totally bounded, some of its subsets still will be.The term precompact (or pre-compact) is sometimes used with the same meaning, but `pre-compact' is also usedto mean relatively compact. In a complete metric space these meanings coincide but in general they do not. See alsouse of the axiom of choice below.

55.1 Definition for a metric space

A metric space (M,d) is totally bounded if and only if for every real number ϵ > 0 , there exists a finite collectionof open balls inM of radius ϵ whose union containsM . Equivalently, the metric spaceM is totally bounded if andonly if for every ϵ > 0 , there exists a finite cover such that the radius of each element of the cover is at most ϵ . Thisis equivalent to the existence of a finite ε-net.[1]

Each totally bounded space is bounded (as the union of finitely many bounded sets is bounded), but the converse isnot true in general. For example, an infinite set equipped with the discrete metric is bounded but not totally bounded.If M is Euclidean space and d is the Euclidean distance, then a subset (with the subspace topology) is totally boundedif and only if it is bounded.A metric space is said to be precompact if every sequence admits a Cauchy subsequence. Thus for metric spaceswe have: compactness = precompactness + completeness. It turns out that the space is precompact if and only if it istotally bounded. Therefore both names can be used interchangeably.

55.2 Definitions in other contexts

The general logical form of the definition is: a subset S of a space X is a totally bounded set if and only if, given anysize E, there exist a natural number n and a family A1, A2, ..., An of subsets of X, such that S is contained in the unionof the family (in other words, the family is a finite cover of S), and such that each set Ai in the family is of size E (orless). In mathematical symbols:

∀E ∃n ∈ N , A1, A2, . . . , An ⊆ X

(S ⊆

n∪i=1

Ai and ∀i = 1, . . . , n size(Ai) ≤ E

).

The space X is a totally bounded space if and only if it is a totally bounded set when considered as a subset of itself.(One can also define totally bounded spaces directly, and then define a set to be totally bounded if and only if it istotally bounded when considered as a subspace.)

240

55.3. EXAMPLES AND NONEXAMPLES 241

The terms “space” and “size” here are vague, and they may be made precise in various ways:A subset S of a metric space X is totally bounded if and only if, given any positive real number E, there exists a finitecover of S by subsets of X whose diameters are all less than E. (In other words, a “size” here is a positive real number,and a subset is of size E if its diameter is less than E.) Equivalently, S is totally bounded if and only if, given any Eas before, there exist elements a1, a2, ..., an of X such that S is contained in the union of the n open balls of radius Earound the points ai.A subset S of a topological vector space, or more generally topological abelian group, X is totally bounded if and onlyif, given any neighbourhood E of the identity (zero) element of X, there exists a finite cover of S by subsets of X eachof which is a translate of a subset of E. (In other words, a “size” here is a neighbourhood of the identity element, anda subset is of size E if it is translate of a subset of E.) Equivalently, S is totally bounded if and only if, given any E asbefore, there exist elements a1, a2, ..., an of X such that S is contained in the union of the n translates of E by thepoints ai.A topological group X is left-totally bounded if and only if it satisfies the definition for topological abelian groupsabove, using left translates. That is, use aiE in place of E + ai. Alternatively, X is right-totally bounded if and only ifit satisfies the definition for topological abelian groups above, using right translates. That is, use Eai in place of E +ai. (In other words, a “size” here is unambiguously a neighbourhood of the identity element, but there are two notionsof whether a set is of a given size: a left notion based on left translation and a right notion based on right translation.)Generalising the above definitions, a subset S of a uniform space X is totally bounded if and only if, given anyentourage E in X, there exists a finite cover of S by subsets of X each of whose Cartesian squares is a subset ofE. (In other words, a “size” here is an entourage, and a subset is of size E if its Cartesian square is a subset of E.)Equivalently, S is totally bounded if and only if, given any E as before, there exist subsets A1, A2, ..., An of X suchthat S is contained in the union of the Ai and, whenever the elements x and y of X both belong to the same set Ai,then (x,y) belongs to E (so that x and y are close as measured by E).The definition can be extended still further, to any category of spaces with a notion of compactness and Cauchycompletion: a space is totally bounded if and only if its completion is compact.

55.3 Examples and nonexamples• A subset of the real line, or more generally of (finite-dimensional) Euclidean space, is totally bounded if andonly if it is bounded. Archimedean property is used.

• The unit ball in a Hilbert space, or more generally in a Banach space, is totally bounded if and only if the spacehas finite dimension.

• Every compact set is totally bounded, whenever the concept is defined.

• Every totally bounded metric space is bounded. However not every bounded metric space is totally bounded.[2]

• A subset of a complete metric space is totally bounded if and only if it is relatively compact (meaning that itsclosure is compact).

• In a locally convex space endowed with the weak topology the precompact sets are exactly the bounded sets.

• A metric space is separable if and only if it is homeomorphic to a totally bounded metric space.[2]

• An infinite metric space with the discrete metric (the distance between any two distinct points is 1) is not totallybounded, even though it is bounded.

55.4 Relationships with compactness and completeness

There is a nice relationship between total boundedness and compactness:Every compact metric space is totally bounded.A uniform space is compact if and only if it is both totally bounded and Cauchy complete. This can be seen as ageneralisation of the Heine–Borel theorem from Euclidean spaces to arbitrary spaces: we must replace boundednesswith total boundedness (and also replace closedness with completeness).

242 CHAPTER 55. TOTALLY BOUNDED SPACE

There is a complementary relationship between total boundedness and the process of Cauchy completion: A uniformspace is totally bounded if and only if its Cauchy completion is totally bounded. (This corresponds to the fact that,in Euclidean spaces, a set is bounded if and only if its closure is bounded.)Combining these theorems, a uniform space is totally bounded if and only if its completion is compact. This maybe taken as an alternative definition of total boundedness. Alternatively, this may be taken as a definition of pre-compactness, while still using a separate definition of total boundedness. Then it becomes a theorem that a space istotally bounded if and only if it is precompact. (Separating the definitions in this way is useful in the absence of theaxiom of choice; see the next section.)

55.5 Use of the axiom of choice

The properties of total boundedness mentioned above rely in part on the axiom of choice. In the absence of theaxiom of choice, total boundedness and precompactness must be distinguished. That is, we define total boundednessin elementary terms but define precompactness in terms of compactness and Cauchy completion. It remains true (thatis, the proof does not require choice) that every precompact space is totally bounded; in other words, if the completionof a space is compact, then that space is totally bounded. But it is no longer true (that is, the proof requires choice)that every totally bounded space is precompact; in other words, the completion of a totally bounded space might notbe compact in the absence of choice.

55.6 See also• Measure of non-compactness

• Locally compact space

55.7 Notes[1] Sutherland p.139

[2] Willard, p. 182

55.8 References• Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.

• Sutherland, W.A. (1975). Introduction to metric and topological spaces. Oxford University Press. ISBN 0-19-853161-3. Zbl 0304.54002.

Chapter 56

Tychonoff’s theorem

For other theorems named after Tychonoff, see Tychonoff’s theorem (disambiguation).

In mathematics, Tychonoff’s theorem states that the product of any collection of compact topological spaces iscompact with respect to the product topology. The theorem is named after Andrey Nikolayevich Tychonoff, whoproved it first in 1930 for powers of the closed unit interval and in 1935 stated the full theorem along with the remarkthat its proof was the same as for the special case. The earliest known published proof is contained in a 1937 paperof Eduard Čech.Several texts identify Tychonoff’s theorem as the single most important result in general topology [e.g. Willard, p.120]; others allow it to share this honor with Urysohn’s lemma.

56.1 Topological definitions

The theorem depends crucially upon the precise definitions of compactness and of the product topology; in fact,Tychonoff’s 1935 paper defines the product topology for the first time. Conversely, part of its importance is to giveconfidence that these particular definitions are the correct (i.e., most useful) ones.Indeed, the Heine–Borel definition of compactness — that every covering of a space by open sets admits a finitesubcovering — is relatively recent. More popular in the 19th and early 20th centuries was the Bolzano–Weierstrasscriterion that every sequence admits a convergent subsequence, now called sequential compactness. These conditionsare equivalent for metrizable spaces, but neither one implies the other in the class of all topological spaces.It is almost trivial to prove that the product of two sequentially compact spaces is sequentially compact — one passesto a subsequence for the first component and then a subsubsequence for the second component. An only slightly moreelaborate “diagonalization” argument establishes the sequential compactness of a countable product of sequentiallycompact spaces. However, the product of continuum many copies of the closed unit interval (with its usual topology)fails to be sequentially compact with respect to the product topology, even though it is compact by Tychonoff’s theorem(e.g., see Wilansky 1970, p. 134).This is a critical failure: if X is a completely regular Hausdorff space, there is a natural embedding from X into[0,1]C(X,[0,1]), where C(X,[0,1]) is the set of continuous maps from X to [0,1]. The compactness of [0,1]C(X,[0,1]) thusshows that every completely regular Hausdorff space embeds in a compact Hausdorff space (or, can be “compacti-fied”.) This construction is the Stone–Čech compactification. Conversely, all subspaces of compact Hausdorff spacesare completely regular Hausdorff, so this characterizes the completely regular Hausdorff spaces as those that can becompactified. Such spaces are now called Tychonoff spaces.

56.2 Applications

Tychonoff’s theorem has been used to prove many other mathematical theorems. These include theorems aboutcompactness of certain spaces such as the Banach–Alaoglu theorem on the weak-* compactness of the unit ball ofthe dual space of a normed vector space, and the Arzelà–Ascoli theorem characterizing the sequences of functions

243

244 CHAPTER 56. TYCHONOFF’S THEOREM

in which every subsequence has a uniformly convergent subsequence. They also include statements less obviouslyrelated to compactness, such as the De Bruijn–Erdős theorem stating that every minimal k-chromatic graph is finite,and the Curtis–Hedlund–Lyndon theorem providing a topological characterization of cellular automata.As a rule of thumb, any sort of construction that takes as input a fairly general object (often of an algebraic, ortopological-algebraic nature) and outputs a compact space is likely to use Tychonoff: e.g., the Gelfand space ofmaximal ideals of a commutative C* algebra, the Stone space of maximal ideals of a Boolean algebra, and theBerkovich spectrum of a commutative Banach ring.

56.3 Proofs of Tychonoff’s theorem

1) Tychonoff’s 1930 proof used the concept of a complete accumulation point.2) The theorem is a quick corollary of the Alexander subbase theorem.More modern proofs have been motivated by the following considerations: the approach to compactness via con-vergence of subsequences leads to a simple and transparent proof in the case of countable index sets. However, theapproach to convergence in a topological space using sequences is sufficient when the space satisfies the first axiomof countability (as metrizable spaces do), but generally not otherwise. However, the product of uncountably manymetrizable spaces, each with at least two points, fails to be first countable. So it is natural to hope that a suitablenotion of convergence in arbitrary spaces will lead to a compactness criterion generalizing sequential compactness inmetrizable spaces that will be as easily applied to deduce the compactness of products. This has turned out to be thecase.3) The theory of convergence via filters, due to Henri Cartan and developed by Bourbaki in 1937, leads to thefollowing criterion: assuming the ultrafilter lemma, a space is compact if and only if each ultrafilter on the spaceconverges. With this in hand, the proof becomes easy: the (filter generated by the) image of an ultrafilter on theproduct space under any projection map is an ultrafilter on the factor space, which therefore converges, to at least onexi. One then shows that the original ultrafilter converges to x = (xi). In his textbook, Munkres gives a reworking ofthe Cartan–Bourbaki proof that does not explicitly use any filter-theoretic language or preliminaries.4) Similarly, the Moore–Smith theory of convergence via nets, as supplemented by Kelley’s notion of a universal net,leads to the criterion that a space is compact if and only if each universal net on the space converges. This criterionleads to a proof (Kelley, 1950) of Tychonoff’s theorem, which is, word for word, identical to the Cartan/Bourbakiproof using filters, save for the repeated substitution of “universal net” for “ultrafilter base”.5) A proof using nets but not universal nets was given in 1992 by Paul Chernoff.

56.4 Tychonoff’s theorem and the axiom of choice

All of the above proofs use the axiom of choice (AC) in some way. For instance, the third proof uses that every filteris contained in an ultrafilter (i.e., a maximal filter), and this is seen by invoking Zorn’s lemma. Zorn’s lemma is alsoused to prove Kelley’s theorem, that every net has a universal subnet. In fact these uses of AC are essential: in 1950Kelley proved that Tychonoff’s theorem implies the axiom of choice. Note that one formulation of AC is that theCartesian product of a family of nonempty sets is nonempty; but since the empty set is most certainly compact, theproof cannot proceed along such straightforward lines. Thus Tychonoff’s theorem joins several other basic theorems(e.g. that every nonzero vector space has a basis) in being equivalent to AC.On the other hand, the statement that every filter is contained in an ultrafilter does not imply AC. Indeed, it is nothard to see that it is equivalent to the Boolean prime ideal theorem (BPI), a well-known intermediate point betweenthe axioms of Zermelo-Fraenkel set theory (ZF) and the ZF theory augmented by the axiom of choice (ZFC). A firstglance at the second proof of Tychnoff may suggest that the proof uses no more than (BPI), in contradiction to theabove. However, the spaces in which every convergent filter has a unique limit are precisely the Hausdorff spaces. Ingeneral we must select, for each element of the index set, an element of the nonempty set of limits of the projectedultrafilter base, and of course this uses AC. However, it also shows that the compactness of the product of compactHausdorff spaces can be proved using (BPI), and in fact the converse also holds. Studying the strength of Tychonoff’stheorem for various restricted classes of spaces is an active area in set-theoretic topology.The analogue of Tychonoff’s theorem in pointless topology does not require any form of the axiom of choice.

56.5. PROOF OF THE AXIOM OF CHOICE FROM TYCHONOFF’S THEOREM 245

56.5 Proof of the axiom of choice from Tychonoff’s theorem

To prove that Tychonoff’s theorem in its general version implies the axiom of choice, we establish that every infinitecartesian product of non-empty sets is nonempty. The trickiest part of the proof is introducing the right topology.The right topology, as it turns out, is the cofinite topology with a small twist. It turns out that every set given thistopology automatically becomes a compact space. Once we have this fact, Tychonoff’s theorem can be applied; wethen use the finite intersection property (FIP) definition of compactness. The proof itself (due to J. L. Kelley) follows:Let Ai be an indexed family of nonempty sets, for i ranging in I (where I is an arbitrary indexing set). We wishto show that the cartesian product of these sets is nonempty. Now, for each i, take Xi to be Ai with the index i itselftacked on (renaming the indices using the disjoint union if necessary, we may assume that i is not a member of Ai,so simply take Xi = Ai ∪ i).Now define the cartesian product

X =∏i∈I

Xi

along with the natural projection maps πi which take a member of X to its ith term.We give each Xi the topology whose open sets are the cofinite subsets of Xi, plus the empty set (the cofinite topology)and the singleton i. This makes Xi compact, and by Tychonoff’s theorem, X is also compact (in the producttopology). The projection maps are continuous; all the Ai's are closed, being complements of the singleton open seti in Xi. So the inverse images πi−1(Ai) are closed subsets of X. We note that

∏i∈I

Ai =∩i∈I

π−1i (Ai)

and prove that these inverse images are nonempty and have the FIP. Let i1, ..., iN be a finite collection of indicesin I. Then the finite product Ai1 × ... × AiN is non-empty (only finitely many choices here, so AC is not needed); itmerely consists of N-tuples. Let a = (a1, ..., aN) be such an N-tuple. We extend a to the whole index set: take a tothe function f defined by f(j) = ak if j = ik, and f(j) = j otherwise. This step is where the addition of the extra point toeach space is crucial, for it allows us to define f for everything outside of the N-tuple in a precise way without choices(we can already choose, by construction, j from Xj ). πik(f) = ak is obviously an element of each Aik so that f is ineach inverse image; thus we have

N∩k=1

π−1ik

(Aik) = ∅.

By the FIP definition of compactness, the entire intersection over I must be nonempty, and the proof is complete.

56.6 References• Chernoff, Paul N. (1992), “A simple proof of Tychonoff’s theorem via nets”, American Mathematical Monthly99 (10): 932–934, doi:10.2307/2324485, JSTOR 2324485.

• Johnstone, Peter T. (1982), Stone spaces, Cambridge Studies in Advanced Mathematics 3, New York: Cam-bridge University Press, ISBN 0-521-23893-5.

• Johnstone, Peter T. (1981), “Tychonoff’s theorem without the axiom of choice”, Fundamenta Mathematica113: 21–35.

• Kelley, John L. (1950), “Convergence in topology”,DukeMathematical Journal 17 (3): 277–283, doi:10.1215/S0012-7094-50-01726-1.

• Kelley, John L. (1950), “The Tychonoff product theorem implies the axiom of choice”, Fundamenta Mathe-matica 37: 75–76.

246 CHAPTER 56. TYCHONOFF’S THEOREM

• Munkres, James (2000), Topology (2nd ed.), Upper Saddle River, NJ: Prentice Hall, ISBN 0-13-181629-2.

• Tychonoff, Andrey N. (1930), "Über die topologische Erweiterung von Räumen”,Mathematische Annalen (inGerman) 102 (1): 544–561, doi:10.1007/BF01782364.

• Wilansky, A. (1970), Topology for Analysis, Ginn and Company

• Willard, Stephen (2004), General Topology, Mineola, NY: Dover Publications, ISBN 0-486-43479-6.

56.7 External links• Tychonoff’s Theorem at ProofWiki

Chapter 57

Union (set theory)

A BA∪B

Union of two sets:A ∪B

In set theory, the union (denoted by ∪) of a collection of sets is the set of all distinct elements in the collection.[1] Itis one of the fundamental operations through which sets can be combined and related to each other.

57.1 Union of two sets

The union of two sets A and B is the set of elements which are in A, in B, or in both A and B. In symbols,

A ∪B = x : x ∈ A or x ∈ B

247

248 CHAPTER 57. UNION (SET THEORY)

Union of three sets:A ∪B ∪ C

For example, if A = 1, 3, 5, 7 and B = 1, 2, 4, 6 then A ∪ B = 1, 2, 3, 4, 5, 6, 7. A more elaborate example(involving two infinite sets) is:

A = x is an even integer larger than 1B = x is an odd integer larger than 1A ∪B = 2, 3, 4, 5, 6, . . .

Sets cannot have duplicate elements, so the union of the sets 1, 2, 3 and 2, 3, 4 is 1, 2, 3, 4. Multipleoccurrences of identical elements have no effect on the cardinality of a set or its contents.The number 9 is not contained in the union of the set of prime numbers 2, 3, 5, 7, 11, … and the set of evennumbers 2, 4, 6, 8, 10, …, because 9 is neither prime nor even.

57.2 Algebraic properties

Binary union is an associative operation; that is,

57.3. FINITE UNIONS 249

A ∪ (B ∪ C) = (A ∪ B) ∪ C.

The operations can be performed in any order, and the parentheses may be omitted without ambiguity (i.e., either ofthe above can be expressed equivalently as A ∪ B ∪ C). Similarly, union is commutative, so the sets can be written inany order.The empty set is an identity element for the operation of union. That is, A ∪ ∅ = A, for any set A.These facts follow from analogous facts about logical disjunction.

57.3 Finite unions

One can take the union of several sets simultaneously. For example, the union of three sets A, B, and C contains allelements of A, all elements of B, and all elements of C, and nothing else. Thus, x is an element of A ∪ B ∪ C if andonly if x is in at least one of A, B, and C.In mathematics a finite union means any union carried out on a finite number of sets: it doesn't imply that the unionset is a finite set.

57.4 Arbitrary unions

The most general notion is the union of an arbitrary collection of sets, sometimes called an infinitary union. IfM isa set whose elements are themselves sets, then x is an element of the union ofM if and only if there is at least oneelement A ofM such that x is an element of A. In symbols:

x ∈∪M ⇐⇒ ∃A ∈M, x ∈ A.

That this union of M is a set no matter how large a set M itself might be, is the content of the axiom of union inaxiomatic set theory.This idea subsumes the preceding sections, in that (for example) A ∪ B ∪ C is the union of the collection A,B,C.Also, if M is the empty collection, then the union of M is the empty set. The analogy between finite unions andlogical disjunction extends to one between arbitrary unions and existential quantification.

57.4.1 Notations

The notation for the general concept can vary considerably. For a finite union of sets S1, S2, S3, . . . , Sn one oftenwrites S1 ∪ S2 ∪ S3 ∪ · · · ∪ Sn . Various common notations for arbitrary unions include

∪M ,

∪A∈MA , and∪

i∈I Ai , the last of which refers to the union of the collection Ai : i ∈ I where I is an index set and Ai is a setfor every i ∈ I . In the case that the index set I is the set of natural numbers, one uses a notation

∪∞i=1Ai analogous

to that of the infinite series. When formatting is difficult, this can also be written "A1 ∪ A2 ∪ A3 ∪ ···". (This lastexample, a union of countably many sets, is very common in analysis; for an example see the article on σ-algebras.)Whenever the symbol "∪" is placed before other symbols instead of between them, it is of a larger size.

57.4.2 Union and intersection

Since sets with unions and intersections form a Boolean algebra, Intersection distributes over union:

A ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C)

and union distributes over intersection:

A ∪ (B ∩ C) = (A ∪B) ∩ (A ∪ C)

250 CHAPTER 57. UNION (SET THEORY)

Within a given universal set, union can be written in terms of the operations of intersection and complement as

A ∪B =(AC ∩BC

)Cwhere the superscript C denotes the complement with respect to the universal set.Arbitrary union and intersection also satisfy the law

∪i∈I

( ∩j∈J

Ai,j

)⊆∩j∈J

(∪i∈I

Ai,j

)

57.5 See also• Alternation (formal language theory), the union of sets of strings

• Cardinality

• Complement (set theory)

• Disjoint union

• Intersection (set theory)

• Iterated binary operation

• Naive set theory

• Symmetric difference

57.6 Notes[1] Weisstein, Eric W. “Union”. Wolfram’s Mathworld. Retrieved 2009-07-14.

57.7 External links• Weisstein, Eric W., “Union”, MathWorld.

• Hazewinkel, Michiel, ed. (2001), “Union of sets”, Encyclopedia ofMathematics, Springer, ISBN 978-1-55608-010-4

• Infinite Union and Intersection at ProvenMath DeMorgan’s laws formally proven from the axioms of set theory.

Chapter 58

σ-compact space

In mathematics, a topological space is said to be σ-compact if it is the union of countably many compact subspaces.[1]

A space is said to be σ-locally compact if it is both σ-compact and locally compact.[2]

58.1 Properties and examples

• Every compact space is σ-compact, and every σ-compact space is Lindelöf (i.e. every open cover has a count-able subcover).[3] The reverse implications do not hold, for example, standard Euclidean space (Rn) is σ-compact but not compact,[4] and the lower limit topology on the real line is Lindelöf but not σ-compact.[5] Infact, the countable complement topology is Lindelöf but neither σ-compact nor locally compact.[6]

• A Hausdorff, Baire space that is also σ-compact, must be locally compact at at least one point.

• If G is a topological group and G is locally compact at one point, then G is locally compact everywhere.Therefore, the previous property tells us that if G is a σ-compact, Hausdorff topological group that is also aBaire space, then G is locally compact. This shows that for Hausdorff topological groups that are also Bairespaces, σ-compactness implies local compactness.

• The previous property implies for instance thatRω is not σ-compact: if it were σ-compact, it would necessarilybe locally compact since Rω is a topological group that is also a Baire space.

• Every hemicompact space is σ-compact.[7] The converse, however, is not true;[8] for example, the space ofrationals, with the usual topology, is σ-compact but not hemicompact.

• The product of a finite number of σ-compact spaces is σ-compact. However the product of an infinite numberof σ-compact spaces may fail to be σ-compact.[9]

• A σ-compact space X is second category (resp. Baire) if and only if the set of points at which is X is locallycompact is nonempty (resp. dense) in X.[10]

58.2 See also

• Exhaustion by compact sets

• Lindelöf space

251

252 CHAPTER 58. Σ-COMPACT SPACE

58.3 Notes[1] Steen, p.19; Willard, p. 126.

[2] Steen, p. 21.

[3] Steen, p. 19.

[4] Steen, p. 56.

[5] Steen, p. 75–76.

[6] Steen, p. 50.

[7] Willard, p. 126.

[8] Willard, p. 126.

[9] Willard, p. 126.

[10] Willard, p. 188.

58.4 References• Steen, Lynn A. and Seebach, J. Arthur Jr.; Counterexamples in Topology, Holt, Rinehart and Winston (1970).ISBN 0-03-079485-4.

• Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.

58.5. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 253

58.5 Text and image sources, contributors, and licenses

58.5.1 Text• A-paracompact space Source: http://en.wikipedia.org/wiki/A-paracompact_space?oldid=532052977 Contributors: Zundark, Paul Au-

gust, Vipul, Michael Slone, SmackBot, Silly rabbit, Lambiam, Cydebot, Ryan, JackSchmidt, Addbot, Brad7777 and Anonymous: 3• Binary relation Source: http://en.wikipedia.org/wiki/Binary_relation?oldid=666434435 Contributors: AxelBoldt, Bryan Derksen, Zun-

dark, Tarquin, Jan Hidders, Roadrunner, Mjb, Tomo, Patrick, Xavic69, Michael Hardy, Wshun, Isomorphic, Dominus, Ixfd64, Takuya-Murata, Charles Matthews, Timwi, Dcoetzee, Jitse Niesen, Robbot, Chocolateboy, MathMartin, Tobias Bergemann, Giftlite, Fropuff,Dratman, Jorge Stolfi, Jlr~enwiki, Andycjp, Quarl, Guanabot, Yuval madar, Slipstream, Paul August, Elwikipedista~enwiki, Shanes,EmilJ, Randall Holmes, Ardric47, Obradovic Goran, Eje211, Alansohn, Dallashan~enwiki, Keenan Pepper, Adrian.benko, Oleg Alexan-drov, Joriki, Linas, MFH, Dpv, Pigcatian, Penumbra2000, Fresheneesz, Chobot, YurikBot, Hairy Dude, Koffieyahoo, Trovatore, Bota47,Arthur Rubin, Netrapt, SmackBot, Royalguard11, SEIBasaurus, Cybercobra, JonAwbrey, Turms, Lambiam, Dbtfz, Mr Stephen,Mets501,Dreftymac, Happy-melon, Petr Matas, CRGreathouse, CBM, Yrodro, WillowW, Xantharius, Thijs!bot, Egriffin, Rlupsa, JAnDbot, MER-C, Magioladitis, Vanish2, David Eppstein, Robin S, Akurn, Adavidb, LajujKej, Owlgorithm, Djjrjr, Policron, DavidCBryant, Quux0r,VolkovBot, Boute, Vipinhari, Anonymous Dissident, PaulTanenbaum, Jackfork, Wykypydya, Dmcq, AlleborgoBot, AHMartin, Ocse-nave, Sftd, Paradoctor, Henry Delforn (old), MiNombreDeGuerra, DuaneLAnderson, Anchor Link Bot, CBM2, Classicalecon, Clue-Bot, Snigbrook, Rhubbarb, Hans Adler, SilvonenBot, BYS2, Plmday, Addbot, LinkFA-Bot, Tide rolls, Jarble, Legobot, Luckas-bot,Yobot, Ht686rg90, Pcap, Labus, Nallimbot, Reindra, FredrikMeyer, AnomieBOT, Floquenbeam, Royote, Hahahaha4, Materialscientist,Belkovich, Citation bot, Racconish, Jellystones, Xqbot, Isheden, Geero, GhalyBot, Ernsts, Howard McCay, Constructive editor, Mark Re-nier, Mfwitten, RandomDSdevel, NearSetAccount, SpaceFlight89, Yunshui, Miracle Pen, Brambleclawx, RjwilmsiBot, Chharvey, Spork-Bot, OnePt618, Sameer143, Socialservice, ResearchRave, ClueBot NG, Wcherowi, Frietjes, Helpful Pixie Bot, Koertefa, ChrisGualtieri,YFdyh-bot, Dexbot, Makecat-bot, Lerutit, Jochen Burghardt, Jodosma, Karim132, Monkbot, Pratincola, , Some1Redirects4Youand Anonymous: 102

• Closed set Source: http://en.wikipedia.org/wiki/Closed_set?oldid=633332819 Contributors: AxelBoldt, Andre Engels, Toby~enwiki,Toby Bartels, Someone else, Patrick, Smelialichu, Michael Hardy, Isomorphic, Dineshjk, Salsa Shark, AugPi, Dcoetzee, Zoicon5, Rob-bot, Tobias Bergemann, Tosha, Giftlite, Ævar Arnfjörð Bjarmason, Lethe, Fropuff, Python eggs, Karl-Henner, PhotoBox, Eep², PaulAugust, Themusicgod1, Cretog8, O18, JohnyDog, Msh210, ABCD, Sligocki, Oleg Alexandrov, Jannex, Isnow, Justin Ormont, MagisterMathematicae, BD2412, Salix alba, Marozols, VKokielov, Chobot, Buggi22, Roboto de Ajvol, RussBot, Grubber, Poulpy, Sardanaphalus,KnightRider~enwiki, SmackBot, Maksim-e~enwiki, JAnDudík, AGeek Tragedy, Dreadstar, Jim.belk, Mets501, J Di, Jackzhp, Jsd, Mya-suda, Thijs!bot, Colin Rowat, Salgueiro~enwiki, Rycee, Ddxc, DragonBot, Josephvk, Me314, Addbot, SomeUsr, Zorrobot, KamikazeBot,Ciphers, NickK, Xqbot, Bdmy, Almabot, Kamixave, RibotBOT, Erik9bot, Stpasha, EmausBot, Irina Gelbukh, DennisIsMe, Wcherowi,Langing, Paolo Lipparini, Charismaa, Marius siuram and Anonymous: 37

• Closure (topology) Source: http://en.wikipedia.org/wiki/Closure_(topology)?oldid=661885869 Contributors: AxelBoldt, Zundark, TheAnome, Toby Bartels, Edemaine, Michael Hardy, Wshun, Andres, Revolver, Charles Matthews, Zoicon5, Grendelkhan, MathMartin,Tobias Bergemann, Tosha, Giftlite, Rich Farmbrough, TedPavlic, Paul August, BenjBot, El C, HasharBot~enwiki, Sligocki, Oleg Alexan-drov, Isnow, Grammarbot, Jshadias, Chobot, BOT-Superzerocool, Banus, Zvika, Selfworm, Melchoir, RuudVisser, Dreadstar, NeilFraser,Gregbard, Salgueiro~enwiki, JAnDbot, Sullivan.t.j, Daniele.tampieri, Policron, Egaida, TXiKiBoT, VVVBot, Reinderien, Veddharta,Outhwest, Thehotelambush, Kumioko, Anchor Link Bot, Addbot, Yobot, Ht686rg90, Calle, Erel Segal, Ciphers, ArthurBot, Xqbot,Erik9bot, DrilBot, EmausBot, ZéroBot, JordiGH, Wcherowi, YFdyh-bot, Tarpuq and Anonymous: 23

• Compact operator Source: http://en.wikipedia.org/wiki/Compact_operator?oldid=664484742 Contributors: Michael Hardy, Takuya-Murata, Charles Matthews, Prumpf, AndrewKepert, Topbanana, Giftlite, BenFrantzDale, Lethe, Lupin, Billlion, LutzL, Oleg Alexan-drov, Linas, Igny, Mathbot, YurikBot, Wavelength, Scineram, Zvika, SmackBot, Silly rabbit, Tsca.bot, Daqu, Tac-Tics, Mct mht, WISo,Eleuther, Salgueiro~enwiki, Sullivan.t.j, STBotD, Kyle the bot, M gol, Jmath666, Biscuittin, Zsuoyb, Addbot, Lightbot, TaBOT-zerem,AnomieBOT, Citation bot, Bdmy, Citation bot 1, John of Reading, Drusus 0, Bomazi, Minimalrho, Mgkrupa, SoSivr and Anonymous:16

• Compact space Source: http://en.wikipedia.org/wiki/Compact_space?oldid=663827307 Contributors: AxelBoldt, Zundark, Andre En-gels, Toby~enwiki, Toby Bartels, Youandme, Patrick, Michael Hardy, Dominus, Firebirth, TakuyaMurata, BenKovitz, Revolver, CharlesMatthews, Dcoetzee, Dysprosia, Jitse Niesen, Zoicon5, Rik Bos, Lumos3, Robbot, MathMartin, Aetheling, Fuelbottle, Tobias Bergemann,Weialawaga~enwiki, Tosha, Giftlite, Markus Krötzsch, Lethe, Fropuff, Dratman, Matt Crypto, Python eggs, DRMacIver, Vivacissama-mente, Pyrop, Rich Farmbrough, TedPavlic, Guanabot, Paul August, El C, Rgdboer, Andi5, Vipul, Teorth, Kjkolb, Dbastos~enwiki,JohnyDog, Eric Kvaalen, Caesura, Oleg Alexandrov, Gmaxwell, Linas, Blair P. Houghton, OdedSchramm, Neocapitalist, Dionyziz, Gra-ham87, Qwertyus, Саша Стефановић, GeoffO, Mike Segal, Mathbot, Sodin, Chobot, Algebraist, YurikBot, Eraserhead1, RussBot,Lenthe, Trovatore, Crasshopper, ManoaChild, Bota47, Ms2ger, Edin1, Eigenlambda, Sardanaphalus, SmackBot, BeteNoir, Incnis Mrsi,Slaniel, Silly rabbit, DHN-bot~enwiki, Tekhnofiend, RFightmaster, Daqu, SashatoBot, Gandalfxviv, Landonproctor, ALife~enwiki, FellCollar, JRSpriggs, Sniffnoy, Sabate, Cydebot, Headbomb, Dbeatty, Futurebird, JAnDbot, Skimnc, Sullivan.t.j, David Eppstein, R'n'B,J.delanoy, Numbo3, Maurice Carbonaro, TomyDuby, Trumpet marietta 45750, Funandtrvl, LokiClock, PMajer, Plclark, Wikimorphism,FMasic, YohanN7, SieBot, JackSchmidt, Deadlyhair, Fakhredinblog, Roed314, Mpd1989, Lartoven, Cenarium, Hans Adler, Lkruijsw,Mathematix, Humanengr, Marc van Leeuwen, SilvonenBot, Algebran, Topology Expert, Lightbot, Matěj Grabovský, Legobot, Luckas-bot, Ht686rg90, Kilom691, Compsonheir, Erel Segal, Ciphers, Citation bot, ArthurBot, Bdmy, Roquedias, Veltas, Point-set topologist,FrescoBot, Sławomir Biały, Citation bot 1, Tkuvho, Rausch, Tim1357, Bhanin, Trappist the monk, 777sms, Thomassteinke, Rjwilmsi-Bot, EmausBot, Dadaist6174, Fly by Night, GoingBatty, Slawekb, Bethnim, AvicBot, Chharvey, D.Lazard, Hanne v, Zstyron, Joel B.Lewis, MerlIwBot, Helpful Pixie Bot, Agemineye, Shivsagardharam, Langing, BG19bot, Paolo Lipparini, TricksterWolf, Perspectiva8,Mureebe, Kuthikuthikuthi, Brirush, Tducote, Mark viking, Mathmensch, Omertamuz and Anonymous: 123

• Compactly embedded Source: http://en.wikipedia.org/wiki/Compactly_embedded?oldid=607163034 Contributors: Toby Bartels, Ben-FrantzDale, Silly rabbit, JAnDbot, Sullivan.t.j, Jmath666, AlleborgoBot, Ulrigo, Dingenis, Addbot, Yobot, Citation bot, Sławomir Biały,Citation bot 1 and Anonymous: 1

• Cover (topology) Source: http://en.wikipedia.org/wiki/Cover_(topology)?oldid=608317742Contributors: Zundark,Matusz, Tobias Berge-mann, Nonick, Fropuff, David Schaich, Paul August, EmilJ, Jumbuck, ABCD, Salix alba, Chobot, Roboto de Ajvol, YurikBot, Jsnx,

254 CHAPTER 58. Σ-COMPACT SPACE

Yliumath, Mhss, Dreadstar, Julian Mendez, Histwr~enwiki, RebelRobot, Policron, VolkovBot, CWii, LokiClock, Synthebot, Enough-said05, Qwfp, Silvercromagnon, SilvonenBot, Addbot, Topology Expert, Ago68, Numbo3-bot, Luckas-bot, Ht686rg90, Ciphers, Xqbot,RibotBOT, FrescoBot, LucienBOT, Petitlait, EmausBot, WikitanvirBot, DeathOfBalance, Noix07 and Anonymous: 24

• Euclidean space Source: http://en.wikipedia.org/wiki/Euclidean_space?oldid=663123399 Contributors: AxelBoldt, Mav, Zundark, Tar-quin, XJaM, Youandme, Tomo, Patrick, Michael Hardy, Dcljr, Karada, Looxix~enwiki, Angela, Charles Matthews, Dysprosia, Gren-delkhan, David Shay, MathMartin, Tobias Bergemann, Tosha, Giftlite, Lethe, Fropuff, Sriehl, DefLog~enwiki, Andycjp, Tomruen,Iantresman, Tzanko Matev, JohnArmagh, Rich Farmbrough, Paul August, Rgdboer, Msh210, Jimmycochrane, PAR, Eddie Dealtry,Dirac1933, Woohookitty, Isnow, Qwertyus, MarSch, MZMcBride, VKokielov, Kolbasz, Fresheneesz, NevilleDNZ, Chobot, Bgwhite,JPD, Wavelength, Hede2000, Epolk, KSmrq, SpuriousQ, ENeville, Mgnbar, Arthur Rubin, Brian Tvedt, RG2, JDspeeder1, SmackBot,Iamhove, Incnis Mrsi, Reedy, Mhss, JoeKearney, Silly rabbit, Hongooi, Tamfang, SashatoBot, Jim.belk, DabMachine, Dan Gluck, Kaarel,Yggdrasil014, Heqs, CmdrObot, GargoyleMT, Rudjek, Philomath3, Aiko, Guy Macon, Orionus, Salgueiro~enwiki, JAnDbot, Bencher-lite, CrizCraig, Magioladitis, TheChard, Avicennasis, Nucleophilic, Oderbolz, R'n'B, Reedy Bot, Policron, Trigamma, The enemies ofgod, Cerberus0, VolkovBot, IWhisky, Philip Trueman, Richardohio, WereSpielChequers, Da Joe, Caltas, Paolo.dL,MiNombreDeGuerra,Lightmouse, Denisarona, Tomas e, Mild Bill Hiccup, Gwguffey, Vsage, DhananSekhar, SilvonenBot, SkyLined, The Rationalist, Addbot,AkhtaBot, Pmod, Tide rolls, Legobot, Yobot, , Collieuk, Materialscientist, Citation bot, Sandip90, Xqbot, St.nerol, Nfr-Maat, Dead-clever23, RoyLeban, Ksuzanne, Mineralquelle, FrescoBot, Sławomir Biały, Alxeedo, RandomDSdevel, Gapato, Mikrosam Akademija2, Yunesj, Wikivictory, EmausBot, John of Reading, Quondum, Gbsrd, ClueBot NG, Wcherowi, Master Uegly, Cntras, Frank.manus,ElectricUvula, ElphiBot, MRG90, FeralOink, Userbot12, Lugia2453, Brirush, Limit-theorem, Eyesnore, Yardimsever, Tentinator, Fen-tonville, Mgkrupa, BemusedObserver, OrganicAltMetal, Ro4sho, Preethambittu, KasparBot and Anonymous: 98

• Exhaustion by compact sets Source: http://en.wikipedia.org/wiki/Exhaustion_by_compact_sets?oldid=379469145 Contributors: Zun-dark, TakuyaMurata, Charles Matthews, Giftlite, Oleg Alexandrov, Catamorphism, SmackBot, Silly rabbit, R'n'B, MikeRumex andAnonymous: 3

• Feebly compact space Source: http://en.wikipedia.org/wiki/Feebly_compact_space?oldid=626310065 Contributors: Zundark, Vipul,Hennobrandsma, Silly rabbit, Vanish2, David Eppstein and Yobot

• Functional analysis Source: http://en.wikipedia.org/wiki/Functional_analysis?oldid=659336520Contributors: AxelBoldt, Zundark, Youandme,Stevertigo, Michael Hardy, Kku, BenKovitz, Rotem Dan, Revolver, Charles Matthews, Dysprosia, Jitse Niesen, Fuzheado, Phys, Bevo,Robbot, Humus sapiens, Tobias Bergemann, Pdenapo,Weialawaga~enwiki, Giftlite, Lethe, Lupin, Fropuff, Ssd, Prosfilaes, DefLog~enwiki,AmarChandra, Abar, Paul August, Bender235, Billlion, Brian0918, Msh210, Derbeth, Oleg Alexandrov, Joriki, Woohookitty, Linas, Igny,Ae-a, Ruud Koot, Grammarbot, Koavf, Mathbot, John Z, YurikBot, Froth, BOT-Superzerocool, Banus, Lunch, Finell, Sardanaphalus,SmackBot, Alsandro, MalafayaBot, Silly rabbit, DHN-bot~enwiki, Can't sleep, clown will eat me, Cwzwarich, Allan McInnes, Merge,Richard L. Peterson, Kevin Murray, CRGreathouse, Yaris678, KennyDC, Thenub314, Magioladitis, RogierBrussee, Althai, Awake-forever, Allstarecho, Jtir, 1000Faces, Policron, VolkovBot, Camrn86, Kyle the bot, Temurjin, TXiKiBoT, Don4of4, Jmath666, Alle-borgoBot, Quietbritishjim, SieBot, Stca74, Brews ohare, DumZiBoT, WikHead, D.M. from Ukraine, Numbo3-bot, Lightbot, Legobot,Luckas-bot, Time Dilation, ArthurBot, Bdmy, Supernova0, Charvest, CES1596, FrescoBot, Kiefer.Wolfowitz, RedBot, Tweet7, SepIHw,Jowa fan, EmausBot, JJasper123, Mathuvw, JFB80, Maxdlink, ClueBot NG, BG19bot, Brad7777, Danwizard208, Randomguess, Kodi-ologist, Brirush, SakeUPenn, Mgkrupa, SoSivr and Anonymous: 85

• H-closed space Source: http://en.wikipedia.org/wiki/H-closed_space?oldid=594778406 Contributors: Paolo Lipparini and Anonymous:2

• Hasse diagram Source: http://en.wikipedia.org/wiki/Hasse_diagram?oldid=641802452 Contributors: Patrick, Michael Hardy, Tim Star-ling, Eric119, Dysprosia, Selket, Hyacinth, Ed g2s, Shantavira, Sbisolo, Altenmann, Bkell, Cyrius, Giftlite, Markus Krötzsch, Guanaco,DefLog~enwiki, Quarl, Noisy, Abar, Rich Farmbrough, Paul August, Kwamikagami, Jmeisen, Mdd, Vanished user zdkjeirj3i46k567,Forejtv~enwiki, Salix alba, Ligulem, Fresheneesz, Chobot, YurikBot, Hairy Dude, Mysid, Closedmouth, SmackBot, Unyoyega, Gelingvis-toj, Mhss, Nbarth, Leland McInnes, Wvbailey, Dreftymac, Zero sharp, CBM, Egriffin, JAnDbot, Deflective, Quentar~enwiki, DavidEppstein, Eybot~enwiki, GaborLajos, Nono le petit robot~enwiki, Anonymous Dissident, PaulTanenbaum, Jamelan, Richtom80, Arcfrk,ClueBot, Watchduck, Addbot, SLWoolf, Luckas-bot, Gilo1969, GrouchoBot, Throw it in the Fire, Citation bot 1, Teuxe, Etincelles,EmausBot, ZéroBot, Matsievsky, ChuispastonBot, Mark viking and Anonymous: 34

• Hausdorff space Source: http://en.wikipedia.org/wiki/Hausdorff_space?oldid=647054341 Contributors: AxelBoldt, Magnus Manske,Zundark, Tarquin, Toby Bartels, B4hand, Michael Hardy, JakeVortex, TakuyaMurata, Ellywa, Cyp, BenKovitz, Ideyal, Charles Matthews,Dcoetzee, Dysprosia, Grendelkhan, Fibonacci, Robbot, MathMartin, Bkell, Tobias Bergemann, Tosha, Giftlite, Fropuff, Icairns, Vasile,Clarknova, TedPavlic, Guanabot, Westendgirl, Paul August, Bender235, El C, Vipul, .:Ajvol:., Tsirel, Dallashan~enwiki, Eric Kvaalen,Sligocki, Isaac, Jim Slim, Oleg Alexandrov, Linas, StradivariusTV, Graham87, BD2412, Chobot, YurikBot, Wavelength, Hairy Dude,Jessesaurus, Bota47, DVDRW, Sardanaphalus, Nbarth, JonAwbrey, Germandemat, Mets501, Dp462090, Cydebot, Alazaris, Dharma6662000,W3asal, Arcresu, RobHar, Salgueiro~enwiki, JAnDbot, AntiSpamBot, Policron, TXiKiBoT, Broadbot, Ocsenave, SieBot, OKBot, Ran-domblue, EconomicsGuy, Kruusamägi, DumZiBoT, Addbot, Mortense, Lightbot, Luckas-bot, Yobot, Ciphers, Citation bot, Kenneth-leebaker, Sławomir Biały, Vectornaut, Lapasotka, EmausBot, Drusus 0, Helpful Pixie Bot, Brad7777, MikeHaskel, Jochen Burghardt,Hierarchivist and Anonymous: 44

• Hemicompact space Source: http://en.wikipedia.org/wiki/Hemicompact_space?oldid=631224234 Contributors: Charles Matthews, To-bias Bergemann, D6, Paul August, Vipul, CommandoGuard~enwiki, Silly rabbit, Cydebot, Vanish2, Addbot, LucienBOT, Mgkrupa andAnonymous: 3

• Interior (topology) Source: http://en.wikipedia.org/wiki/Interior_(topology)?oldid=659062423 Contributors: Mav, The Anome, TobyBartels, TakuyaMurata, GTBacchus, Revolver, Charles Matthews, Dysprosia, Tosha, Giftlite, Jason Quinn, Yuval madar, EmilJ, Ole-galexandrov, Oleg Alexandrov, Linas, Isnow, Mike Segal, Juan Marquez, Margosbot~enwiki, RexNL, Kri, Chobot, Roboto de Ajvol,YurikBot, Splash, Gwaihir, Pred, Poulpy, Banus, Zvika, Selfworm, BiT, JAn Dudík, Bluebot, Dreadstar, Mwtoews, Lambiam, Digana,Madmath789, Vaughan Pratt, Iokseng, JAnDbot, Mathematrucker, SieBot, Thehotelambush, Anchor Link Bot, Carolus m, BOTarate,Addbot, CarsracBot, Luckas-bot, TaBOT-zerem, Erel Segal, Ciphers, MorphismOfDoom, Zfeinst, Chewings72, Stephan Kulla, Mgkrupaand Anonymous: 24

• K-cell (mathematics) Source: http://en.wikipedia.org/wiki/K-cell_(mathematics)?oldid=648451922 Contributors: Michael Hardy, JoeDecker, Gilliam, KathrynLybarger, Rankersbo, Alvin Seville, Paulschn, Brirush, Ncandido, Nkrish96 and Anonymous: 3

58.5. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 255

• Lebesgue covering dimension Source: http://en.wikipedia.org/wiki/Lebesgue_covering_dimension?oldid=660515923Contributors: Ax-elBoldt, Zundark, The Anome, Charles Matthews, MathMartin, Tobias Bergemann, Tosha, Jason Quinn, Eequor, Kuratowski’s Ghost,Cogent, DiegoMoya, OlegAlexandrov, Joriki, Linas, OdedSchramm,Mathbot, YurikBot, Trovatore, RL0919, SmackBot, Mohan1986, AGeek Tragedy, Kuru, Noegenesis, JMK, Mulder416sBot, CBM, Mct mht, VictorAnyakin, Hut 8.5, David Eppstein, Policron, Sigmundur,Izno, TXiKiBoT, PipepBot, Grubb257, Alexbot, 1ForTheMoney, DumZiBoT, Addbot, 5 albert square, Luckas-bot, AnomieBOT,Arthur-Bot, Xqbot, GrouchoBot, SassoBot, AllCluesKey, MondalorBot, ZéroBot, BG19bot, Mark viking and Anonymous: 21

• Limit point compact Source: http://en.wikipedia.org/wiki/Limit_point_compact?oldid=544713924 Contributors: Paul August, Linas,Algebraist, Silly rabbit, Gandalfxviv, Myasuda, Perturbationist, Addbot, Topology Expert, Semistablesystem, 777sms, WikitanvirBot,Paolo Lipparini and Anonymous: 3

• Lindelöf space Source: http://en.wikipedia.org/wiki/Lindel%C3%B6f_space?oldid=643741409 Contributors: Zundark, Michael Hardy,Dominus, Loren Rosen, Revolver, Lumos3, Robinh, Tobias Bergemann, Tosha, Fropuff, Paul August, BenjBot, Vipul, Burn, Linas, R.e.b.,YurikBot, Hairy Dude, Hennobrandsma,Mysid, Bota47, Kompik, SmackBot, OdMishehu, Silly rabbit, Vina-iwbot~enwiki, Stotr~enwiki,Cydebot, Nadav1, Sullivan.t.j, David Eppstein, Marcosaedro, Popopp, JackSchmidt, Andrewbt, DragonBot, MystBot, Addbot, Cuaxdon,LaaknorBot, Ginosbot, Yobot, Citation bot, Xqbot, BenzolBot, Citation bot 1, RedBot, EmausBot, WikitanvirBot, Slawekb, ZéroBot,CitationCleanerBot, Brad7777, YFdyh-bot, Hamoudafg, K9re11, Forgetfulfunctor00, Zdell271 and Anonymous: 15

• Locally compact space Source: http://en.wikipedia.org/wiki/Locally_compact_space?oldid=666366977 Contributors: AxelBoldt, Zun-dark, Toby~enwiki, Toby Bartels, Michael Hardy, Charles Matthews, Jitse Niesen, Lumos3, Shantavira, Robbot, MathMartin, Tosha,Giftlite, Dbenbenn, K igor k, Lupin, Fropuff, Paul August, Vipul, Oleg Alexandrov, BD2412, Jshadias, FlaBot, Mathbot, John Z, Yurik-Bot, Hairy Dude, SmackBot, Andy M. Wang, Bluebot, Silly rabbit, A Geek Tragedy, HLwiKi, Gala.martin, Danpovey, Gandalfxviv,Mets501, Myasuda, Mct mht, Equendil, Cydebot, Sagaciousuk, Headbomb, GurchBot, R'n'B, Ale2006, Marcosaedro, Wikimorphism,JackSchmidt, Addbot, Ginosbot, Luckas-bot, Amirobot, Citation bot, Xqbot, מדר ,יובל Citation bot 1, I dream of horses, EmausBot,PatrickR2, ChrisGualtieri, Gmkwo, Hymath, K9re11, Cohomology84 and Anonymous: 28

• Locally finite Source: http://en.wikipedia.org/wiki/Locally_finite?oldid=572023546 Contributors: Zundark, Revolver, Melchoir, CBM,Addbot, Yaddie, ZéroBot and Anonymous: 1

• Locally finite collection Source: http://en.wikipedia.org/wiki/Locally_finite_collection?oldid=661263548Contributors: Zundark,MichaelHardy, Jitse Niesen, Tobias Bergemann, Giftlite, OdedSchramm, RussBot, Archelon, SmackBot, Silly rabbit, Lambiam, Konradek, R'n'B,JackSchmidt, Sun Creator, Addbot, Topology Expert, Yobot, Ht686rg90, Point-set topologist, ZéroBot, PatrickR2, Helpful Pixie Bot,Pratyush Sarkar and Anonymous: 6

• Locally finite space Source: http://en.wikipedia.org/wiki/Locally_finite_space?oldid=572144183 Contributors: Melchoir, Silly rabbitand David Eppstein

• Manifold Source: http://en.wikipedia.org/wiki/Manifold?oldid=667192868Contributors: Zundark, TheAnome, XJaM, Edemaine,MichaelHardy, Alodyne, Ciphergoth, Rmilson, Charles Matthews, Timwi, Joshuabowman, Jitse Niesen, Mtcv, Banno, Catskul, Fredrik, Sver-drup, Timrollpickering, Tobias Bergemann, Tosha, Giftlite, BenFrantzDale, Lethe, Anville, Jason Quinn, Python eggs, Bobblewik, Me-likamp, Pmanderson, Elroch, TzankoMatev, Smimram, Cacycle, Sperling, Paul August, Ben Standeven, Gauge, El C, Szquirrel, Bobo192,Kevin Lamoreau, Quasicharacter, Obradovic Goran, Haham hanuka, Crust, Varuna, Schissel, Alansohn, Anthony Appleyard, Cdc, OlegAlexandrov, Optimusnauta, Joriki, Linas, Daniel Case, Ruud Koot, Joke137, Mandarax, BD2412, Chun-hian, NatusRoma, OneWeird-Dude, MarSch, Salix alba, NeonMerlin, Juan Marquez, R.e.b., Quuxplusone, DaGizza, Krishnavedala, Bgwhite, Algebraist, Wavelength,Gene.arboit, Loom91, Markus Schmaus, KSmrq, Rick Norwood, Tong~enwiki, Dethomas, Vb, Voidxor, Hv, Acer, Pred, Marlasdad,Jaysbro, Zvika, SmackBot, Nihonjoe, Incnis Mrsi, Slashme, Gcdart, Wisygig, Jjalexand, Silly rabbit, StrangerInParadise, SEIBasaurus,Nbarth, Sisodia, AdamSmithee, Foxjwill, Chlewbot, Berland, QFT, HLwiKi, Leland McInnes, Jdlambert, Jon Awbrey, Acdx, Arglebar-gleIV, Loodog, Loadmaster, Adriferr, EmreDuran, Newone, James pic, Ranicki, Gregbard, Vanished user fj0390923roktg4tlkm2pkd,Ntsimp, Corpx, Madmonk325, Invisible Capybara, Pascal.Tesson, Amitushtush, Msnicki, Xtv, Gimmetrow, Bioguy, Thijs!bot, Anti-VandalBot, Exteray, Dougher, JAnDbot, Michael Tiemann, Norailyain, YK Times, JamesBWatson, Jakob.scholbach, Maniwar, DavidEppstein, Martynas Patasius, Renetus, R'n'B, Pbroks13, J.delanoy, Maurice Carbonaro, Lantonov, Sormani, Policron, Vanished user47736712, LeighvsOptimvsMaximvs, Sarregouset, Camrn86, JohnBlackburne, LokiClock, Bovineboy2008, Aesopos, Greclevoir, Anony-mous Dissident, Appoose, Geometry guy, Sploonie, Arcfrk, Bikasuishin, Katzmik, Rybu, GirasoleDE, Stca74, WereSpielChequers,Domination989, Squelle, Soler97, JackSchmidt, Randomblue, Jludwig, Sidiropo, LarRan, Mr. Granger, Loren.wilton, Gopalkrishnan83,ClueBot, Plastikspork, JuPitEer, Sun Creator, Invive, Brews ohare, Eloifigueiredo, AnonyScientist, XLinkBot, Bradv, Luca Antonelli,CàlculIntegral, Addbot, Topology Expert, MrVanBot, LinkFA-Bot, Lightbot, ,سعی Yobot, Ht686rg90, AnomieBOT, VanishedUsersdu9aya9fasdsopa, Citation bot, Gsmgm, Expooz, TinucherianBot II, Tasudrty, Adcoon, Philipp Kuehl, Cpryby, Br77rino, Point-settopologist, FrescoBot, AllCluesKey, Sławomir Biały, Lost-n-translation, Tkuvho, Pmokeefe, RedBot, Rausch, Gryllida, FoxBot, Ac-tuallyRationalThinker, Bj norge, Jowa fan, WildBot, EmausBot, Slawekb, ZéroBot, Moorechen, Quondum, D.Lazard, Patatas101, AliDadsetan, Anita5192, Mgvongoeden, Helpful Pixie Bot, Tub82911, AvocatoBot, Paweł Ziemian, Agent Swipe, Dexbot, BeaumontTaz,Mark viking, Purnendu Karmakar, Samreid94, Pwm86, K9re11, Rcehy, BethNaught, Karandodia, UtherSB, KasparBot and Anonymous:161

• Mathematical analysis Source: http://en.wikipedia.org/wiki/Mathematical_analysis?oldid=667027391 Contributors: AxelBoldt, LeeDaniel Crocker, Tarquin, Miguel~enwiki, Peterlin~enwiki, Ben-Zin~enwiki, Youandme, Michael Hardy, Wshun, Norm, Iulianu, Snoyes,Andrewa, Cyan, Charles Matthews, Dino, Dysprosia, Tpbradbury, Traroth, Robbot, Fredrik, Romanm, Gandalf61, MathMartin, Fu-elbottle, Tobias Bergemann, Snobot, Giftlite, Lethe, Dratman, Sam Hocevar, PhotoBox, D6, HedgeHog, Urvabara, Paul August, Ben-der235, Tompw, Art LaPella, Nk, Mdd, Msh210, Alansohn, Dallashan~enwiki, Sligocki, Olegalexandrov, Almafeta, Oleg Alexandrov,Linas, Igny, Mandarax, Rjwilmsi, Mayumashu, Koavf, MarSch, Juan Marquez, FlaBot, JYOuyang, Otets, Malhonen, Chobot, DVdm,Borgx, Spacepotato, Hairy Dude, Deeptrivia, RussBot, KSmrq, Chaos, Amplimax, MaNeMeBasat, Pred, Ilmari Karonen, Lunch, Finell,Sardanaphalus, SmackBot, Selfworm, Lestrade, InverseHypercube, Melchoir, Bomac, Jagged 85, Alsandro, Grokmoo, SMP, DarthPanda, Vanished User 0001, SundarBot, LkNsngth, Stefano85, Bidabadi~enwiki, SashatoBot, Lambiam, Ckatz, Daphne A, Aeternus,CRGreathouse, CBM, Thomasmeeks, FilipeS, Rifleman 82, M a s, Omicronpersei8, Cj67, Urdutext, Escarbot, AntiVandalBot, LunaSantin, JAnDbot, MER-C, Thenub314, Yill577, Hurmata, Kuyabribri, KhalidMahmood, Jtir, R'n'B, ZRV, J.delanoy, Maurice Carbonaro,Jonathanzung, Jwuthe2, KCinDC, Juliancolton, DavidCBryant, Treisijs, GregWoodhouse, Useight, Funandtrvl, VolkovBot, JohnBlack-burne, Greclevoir, Altruism, Rei-bot, BotKung, Falcon8765, Thric3, Symane, SieBot, Neworder1, Lagrange613, Zedlik, DesolateReality,Altzinn, Smithpith, Cenarium, BOTarate, Kruusamägi, XLinkBot, SilvonenBot, JinJian, ElMeBot, D.M. from Ukraine, Leonini, Ad-dbot, Friginator, LaaknorBot, CarsracBot, Ozob, Legobot, Luckas-bot, Yobot, Ht686rg90, Quangbao, 9258fahsflkh917fas, Wrelwser43,

256 CHAPTER 58. Σ-COMPACT SPACE

ArthurBot, Xqbot, Dowjgyta, Txebixev, Psyoptix, Gaussy, GrouchoBot, Point-set topologist, RibotBOT, Charvest, Geoffreybernardo,Tkuvho, Hard Sin, RedBot, Allen 6666, FoxBot, EmausBot, WikitanvirBot, Slawekb, Bethnim, QuentinUK, Wayne Slam, Lorem Ip,Herebo, ClueBot NG, Rurik the Varangian, Helpful Pixie Bot, Bcapetta, Alelbre, BG19bot, Mistory, Artem Karimov, Huntingg, Hill-crest98, Brad7777, Christian Glodzinski, Webclient101, Jcardazzi, Brirush, Dave Bowman - Discovery Won, Limit-theorem, K401sTL3,Monkbot, Raghav statistics jaipur, Degenerate prodigy and Anonymous: 117

• Mesocompact space Source: http://en.wikipedia.org/wiki/Mesocompact_space?oldid=544373876 Contributors: Zundark, Cyde, Vipul,Linas, SmackBot, Silly rabbit, Cydebot, Thijs!bot, Addbot and Anonymous: 1

• Metacompact space Source: http://en.wikipedia.org/wiki/Metacompact_space?oldid=662330601 Contributors: Zundark, Tobias Berge-mann, Fropuff, Vipul, Linas, Awis, Rjwilmsi, Algebraist, Silly rabbit, CBM, Cydebot, Ntsimp, Vanish2, JackSchmidt, Addbot, TopologyExpert, Luckas-bot, Citation bot 1, Trappist the monk, Schojoha and Anonymous: 2

• Metric space Source: http://en.wikipedia.org/wiki/Metric_space?oldid=666713106 Contributors: AxelBoldt, LC~enwiki, Zundark, Tar-quin, XJaM, Toby Bartels, Edemaine, Paul Ebermann, Tomo, Patrick, Chas zzz brown, Michael Hardy, Wshun, SGBailey, TakuyaMurata,Looxix~enwiki, Andres, Tristanb, Ideyal, Revolver, RodC, Charles Matthews, Dcoetzee, Dfeuer, Dysprosia, Jitse Niesen, Prumpf, Saltine,AndrewKepert, Mtcv, AnanthaRaman, Donarreiskoffer, Robbot, RedWolf, Altenmann, Romanm, MathMartin, Robinh, Aetheling, To-bias Bergemann, Tosha, Giftlite, Bob Palin, Gene Ward Smith, Markus Krötzsch, Lupin, Fropuff, David Johnson, Python eggs, Gubbubu,LiDaobing, Vivacissamamente, GamerRemag12@aol.com, Pyrop, TedPavlic, Guanabot, Paul August, SpookyMulder, BACbKA, Kinita-wowi, El C, Rgdboer, Crisófilax, Miraage, Blotwell, Emhoo~enwiki, Obradovic Goran, Helix84, Tsirel, Jumbuck, Eric Kvaalen, Sligocki,Fiedorow, Themillofkeytone, Pashi, Kbolino, Oleg Alexandrov, Saeed, Joriki, Linas, Tabletop, Plowboylifestyle, Nileshbansal, Marudub-shinki, Graham87, BD2412, Salix alba, Brighterorange, Mathbot, Margosbot~enwiki, Vulturejoe, Jenny Harrison, CiaPan, Chobot,YurikBot, Hairy Dude, Calumny, Number 57, Stefan Udrea, Kompik, Nothlit, JahJah, Pred, MullerHolk, RonnieBrown, Sardanaphalus,SmackBot, Incnis Mrsi, Hammerite, PJTraill, Complexica, Nbarth, Hve, Xyzzy n, Meni Rosenfeld, Hyperwired, Mets501, Madmath789,Buckyboy314, Roland.barrat, CRGreathouse, KerryVeenstra, CmdrObot, Jackzhp, Thijs!bot, Epbr123, Rlupsa, Seanskye, Urdutext,Orionus, Dougher, JAnDbot, MER-C, Quentar~enwiki, Douglas Whitaker, Extropian314, Magioladitis, Wlod, JJ Harrison, David Epp-stein, Oravec, MartinBot, TheSeven, Policron, Undernearththeman, GSpeight, LokiClock, Moswiki, TXiKiBoT, Plclark, Wikimorphism,Chirpstation, CenturionZ 1, Psymun747, SieBot, YonaBot, WereSpielChequers, Garde, Paolo.dL, MiNombreDeGuerra, JorgenW, Skep-tical scientist, Anchor Link Bot, Melcombe, Cliff, UKoch, Lbertolotti, Hans Adler, Vanished user tj23rpoij4tikkd, DumZiBoT, XLinkBot,Charles Sturm, Addbot, Tjlaxs, Haruth, LaaknorBot, Dyaa, Okcash, Zorrobot, Luckas-bot, Yobot, Ciphers, Joule36e5, Bdmy, Syena,Defeng.wu, DrilBot, HRoestBot, Kiefer.Wolfowitz, Sra2114, SpaceFlight89, Lotje, Jesse V., Biker333, Bethnim, Josve05a, Chharvey,SporkBot, L Kensington, ResearchRave, Reineke80, Wcherowi, Kstouras, Lifeonahilltop, Vinícius Machado Vogt, Helpful Pixie Bot,Tasky2, AdventurousSquirrel, Brad7777, Darvii, Lolmid, Teddyktchan, Verdana Bold, Harryalerta, KasparBot and Anonymous: 101

• Metrization theorem Source: http://en.wikipedia.org/wiki/Metrization_theorem?oldid=660467346 Contributors: AxelBoldt, Zundark,Koyaanis Qatsi, Michael Hardy, Gabbe, TakuyaMurata, Mark Foskey, MathMartin, Tosha, Giftlite, Dbenbenn, El C, Rgdboer, Linas,Sodin, YurikBot, Trovatore, Hennobrandsma, SmackBot, BeteNoir, Reedy, CBM, Gregbard, Thijs!bot, Weixifan, KennyDC, Klein-Klio~enwiki, Jwuthe2, Mszudzik, Stca74, Alexbot, WikHead, Addbot, Topology Expert, Luckas-bot, Xqbot, FrescoBot, LucienBOT,777sms, Ripchip Bot, EmausBot, Immunize, Bengski68 and Anonymous: 18

• Normal space Source: http://en.wikipedia.org/wiki/Normal_space?oldid=626354394 Contributors: AxelBoldt, Toby~enwiki, Toby Bar-tels, Michael Hardy, Dominus, Revolver, Jitse Niesen, Fibonacci, Robbot, Tobias Bergemann, Weialawaga~enwiki, Tosha, Giftlite,Fropuff, Waltpohl, DefLog~enwiki, Vipul, Oleg Alexandrov, Marudubshinki, Mathbot, YurikBot, Hairy Dude, Hennobrandsma, Sar-danaphalus, SmackBot, Mhss, Bluebot, Germandemat, Stotr~enwiki, CRGreathouse, Myasuda, Cydebot, Ntsimp, KennyDC, Jay Gatsby,David Eppstein, JadeNB, Adavidb, Don4of4, Arcfrk, SilvonenBot, Addbot, Topology Expert, RobertHannah89, Luckas-bot, Ptbotgourou,Calle, Citation bot, GrouchoBot, Jonesey95, Rickhev1, RjwilmsiBot, D.Lazard, BG19bot, Paolo Lipparini, Brad7777, Mgkrupa andAnonymous: 13

• Open set Source: http://en.wikipedia.org/wiki/Open_set?oldid=664841828Contributors: AxelBoldt, Zundark, Taral, XJaM, Toby~enwiki,Toby Bartels, Miguel~enwiki, Patrick, Iulianu, Iorsh, Rob Hooft, Dino, Wikiborg, Dysprosia, Jitse Niesen, Robbot, MathMartin, Bkell,Saforrest, Tobias Bergemann, Giftlite, Lethe, Dratman, Jason Quinn, Python eggs, LiDaobing, Tzanko Matev, Moxfyre, Paul August,Elwikipedista~enwiki, Nickj, O18, Obradovic Goran, Nsaa, Gerweck, Dethron, Oleg Alexandrov, Simetrical, Mpatel, Magister Math-ematicae, Salix alba, Mike Segal, FlaBot, VKokielov, Margosbot~enwiki, Jrtayloriv, Chobot, Roboto de Ajvol, YurikBot, Hairy Dude,Bota47, Icedwater, Pred, Paul D. Anderson, Allens, Sardanaphalus, SmackBot, Incnis Mrsi, K-UNIT, Foxjwill, HLwiKi, Danpovey, LkN-sngth, Wizardman, Vina-iwbot~enwiki, Vanished user 9i39j3, AutomaticWriting, Jackzhp, Dslc, Xantharius, Dharma6662000, Thijs!bot,Tchakra, JAnDbot, Richard Giuly, Hashem62, Joeabauer, TomyDuby, Trumpet marietta 45750, Daniele.tampieri, Daniel5Ko, Policron,Dubhe.sk, Chrystomath, KylieTastic, VolkovBot, Camrn86, Cbigorgne, Jordankayla123, MartinPackerIBM, Ocsenave, SieBot, MiNom-breDeGuerra, Thobitz, Blacklemon67, Estirabot, Qwfp, Addbot, Wikomidia, Zorrobot, Legobot, EdwardLane, AnomieBOT, Ciphers,Jim1138, DannyAsher, Bdmy, Point-set topologist, RibotBOT, LivingBot, MarcelB612, Sheerun, Wham Bam Rock II, Grondilu, Erithil,ChuispastonBot, Mesoderm, MRG90, ChalkboardCowboy, Brirush, Horsegal1, Ganatuiyop and Anonymous: 58

• Order theory Source: http://en.wikipedia.org/wiki/Order_theory?oldid=666554129 Contributors: Bryan Derksen, Toby Bartels, MichaelHardy, Dineshjk, Ehn, Charles Matthews, Dcoetzee, Jitse Niesen, Wik, Natevw, VeryVerily, Populus, Topbanana, Robbot, Henrygb, El-Benevolente, Tobias Bergemann, Giftlite, Markus Krötzsch, Elias, DefLog~enwiki, Yarnover, APH, SimonLyall, Xrchz, Abar, Pjacobi,Paul August, Tompw, Nickj, Themusicgod1, Lysdexia, Arthena, Joriki, Linas, Jeff3000, Josh Parris, Salix alba, Mathbot, Hairy Dude,Dmharvey, Trovatore, Ott2, Arthur Rubin, Modify, Netrapt, That Guy, From That Show!, SmackBot, KnowledgeOfSelf, Mhss, Blue-bot, RDBrown, Cybercobra, Kntrabssi, Dreadstar, JohnI, 16@r, Loadmaster, Dicklyon, Landonproctor, Levineps, Dreftymac, Majora4,CRGreathouse, CBM, Sam Staton, Skittleys, Ankit mcgill, MER-C, VoABot II, David Eppstein, Gwern, Maurice Carbonaro, Inquam,Daniel5Ko, Jorfer, JohnBlackburne, Trondarild, Philip Trueman, GcSwRhIc, The Tetrast, Magmi, Geometry guy, Tomaxer, Steven-Johnston, AS, Anchor Link Bot, Randomblue, ClueBot, Justin W Smith, Hans Adler, Wikidsp, Addbot, Barak Sh, Badou517, Legobot,Buenasdiaz, Smallman12q, Tuetschek, FrescoBot, Mark Renier, Orhanghazi, Chenopodiaceous, Gamewizard71, Genezistan, John ofReading, Dadaist6174, ClueBot NG, Syamino, MerlIwBot, Helpful Pixie Bot, Knwlgc, VolunBute, Brad7777, Nicuchalan1, K401sTL3,JaconaFrere, Srlgator, Pheello87, KasparBot and Anonymous: 59

• Orthocompact space Source: http://en.wikipedia.org/wiki/Orthocompact_space?oldid=544373090Contributors: Zundark, Fropuff, Vipul,Linas, Rjwilmsi, Hennobrandsma, SmackBot, Silly rabbit, CMG, Cydebot, Vanish2, Addbot, Amirobot, Andytoh and Helpful Pixie Bot

• Paracompact space Source: http://en.wikipedia.org/wiki/Paracompact_space?oldid=660911615Contributors: AxelBoldt, Zundark, Toby~enwiki,TobyBartels, Michael Hardy, Revolver, CharlesMatthews, Dfeuer, Dysprosia, JitseNiesen, Fibonacci, Tobias Bergemann,Weialawaga~enwiki,

58.5. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 257

Tosha, Giftlite, Lethe, Fropuff, Brockert, Paul August, Vipul, Kuratowski’s Ghost, Don Reba, ABCD, Oleg Alexandrov, OdedSchramm,Jshadias, Salix alba, R.e.b., Mathbot, Trovatore, Hennobrandsma, SmackBot, Navilluskram, Silly rabbit, Dreadstar, Turms, Mets501,Stotr~enwiki, CBM, Cydebot, Thijs!bot, Headbomb, Vanish2, Jakob.scholbach, David Eppstein, Policron, Hqb, Marcosaedro, Mr.Axiom,MikeRumex, Quietbritishjim,Mrw7,Megaloxantha, Grubb257, AlexeyMuranov, Protony~enwiki, Addbot, Topology Expert, Dingo1729,,ماني Yobot, Kilom691, Hank hu, Citation bot, Txebixev, Howard McCay, LucienBOT, Lost-n-translation, Jonesey95, DixonDBot,Dinamik-bot, EmausBot, Slawekb, Drusus 0, Anselrill, TobiTobsensWiki, Brad7777, Dexbot, Mark viking, Mat.wyszynski and Anony-mous: 29

• Partially ordered set Source: http://en.wikipedia.org/wiki/Partially_ordered_set?oldid=663537797 Contributors: Bryan Derksen, Zun-dark, Tomo, Patrick, Bcrowell, Chinju, TakuyaMurata, GTBacchus, AugPi, Charles Matthews, Timwi, Dcoetzee, Dysprosia, Doradus,Maximus Rex, Fibonacci, Tobias Bergemann, Giftlite, Markus Krötzsch, Fropuff, Peruvianllama, Jason Quinn, Neilc, Gubbubu, De-fLog~enwiki, MarkSweep, Urhixidur, TheJames, Paul August, Zaslav, Spoon!, Porton, Haham hanuka, DougOrleans, Msh210, OlegAlexandrov, Daira Hopwood, MFH, Salix alba, FlaBot, Vonkje, Chobot, Laurentius, Dmharvey, Vecter, Sanguinity, Modify, RDBury,Incnis Mrsi, Brick Thrower, Cesine, Zanetu, Jcarroll, Nbarth, Jdthood, Javalenok, Kjetil1001, Dreadstar, Esoth~enwiki, Mike Fikes, A.Pichler, Vaughan Pratt, CRGreathouse, L'œuf, CBM, Werratal, Rlupsa, CZeke, Ill logic, JAnDbot, MER-C, BrotherE, Tbleher, A3nm,David Eppstein, SlamDiego, Bissinger, Haseldon, Daniel5Ko, GaborLajos, NewEnglandYankee, Orphic, RobertDanielEmerson, TXiK-iBoT, Digby Tantrum, PaulTanenbaum, Arcfrk, SieBot, Mochan Shrestha, TheGhostOfAdrianMineha, Thehotelambush, Megaloxantha,Peiresc~enwiki, Cheesefondue, Jludwig, ClueBot, Morinus, Justin W Smith, Methossant, Pi zero, Jonathanrcoxhead, Watchduck, Com-puterGeezer, He7d3r, Hans Adler, Jtle515, Palnot, Marc van Leeuwen, Ankan babee, Addbot, Download, Luckyz, Legobot, Kilom691,AnomieBOT, Erel Segal, Citation bot, SteveWoolf, Undsoweiter, FrescoBot, Nicolas Perrault III, Confluente, Ricardo Ferreira de Oliveira,Throw it in the Fire, Gnathan87, Setitup, EmausBot, John of Reading, Febuiles, Thecheesykid, ZéroBot, Chharvey, The man who wasFriday, SporkBot, Zfeinst, Rathgemz, CocuBot, Vdamanafshan, Mesoderm, MerlIwBot, Wbm1058, Jakshap, Paolo Lipparini, ElphiBot,Larion Garaczi, Aabhis, Jochen Burghardt, Mark viking, Eamonford, Sgbmyr, K401sTL3, Tudor987, Victor Lesyk, Some1Redirects4Youand Anonymous: 77

• Partition of unity Source: http://en.wikipedia.org/wiki/Partition_of_unity?oldid=627688008 Contributors: Mav, Toby~enwiki, Dys-prosia, Giftlite, BenFrantzDale, Lethe, Fropuff, Dratman, Jorge Stolfi, ArnoldReinhold, Oleg Alexandrov, MFH, YurikBot, Trovatore,SmackBot, MisterHand, Mhym, Dreadstar, N2e, Headbomb, Albmont, Jakob.scholbach, Joedalion, Marcosaedro, YouRang?, Myst-Bot, Addbot, Topology Expert, Numbo3-bot, Ht686rg90, Txebixev, Point-set topologist, Erik9bot, Rausch, Mickeyjetson428, Boriaj,Khazar2, Denysbondar and Anonymous: 10

• Product topology Source: http://en.wikipedia.org/wiki/Product_topology?oldid=665576791 Contributors: AxelBoldt, Zundark, MichaelHardy, Karada, Revolver, Dcoetzee, Dysprosia, Zoicon5, Robbot, Rvollmert, Altenmann, Tosha, Giftlite, Fropuff, Waltpohl, JasonQuinn, TedPavlic, El C, O18, Oleg Alexandrov, Linas, Ryan Reich, BD2412, MarSch, Chobot, YurikBot, Michael Slone, Tong~enwiki,Welsh, Banus, Melchoir, Nbarth, Henning Makholm, Tawkerbot2, Dr. Poret, .anacondabot, Albmont, Huzzlet the bot, LordAnubisBOT,VolkovBot, Kyle the bot, Jmath666, Mike4ty4, Arcfrk, Qwfp, DumZiBoT, Leen Droogendijk, Legobot, Luckas-bot, Yobot, Ptbotgourou,Xqbot, FrescoBot, Sławomir Biały, Lapasotka, Reach Out to the Truth, Fabiangabel and Anonymous: 30

• Pseudocompact space Source: http://en.wikipedia.org/wiki/Pseudocompact_space?oldid=622051894 Contributors: Fropuff, Elroch,Vipul, Oleg Alexandrov, Linas, Rjwilmsi, Algebraist, Bluebot, Silly rabbit, Vaughan Pratt, Jac16888, Cydebot, Sullivan.t.j, David Epp-stein, SieBot, Addbot, Topology Expert, Luckas-bot, Yobot, Citation bot 1, RjwilmsiBot, Schojoha and Anonymous: 3

• Realcompact space Source: http://en.wikipedia.org/wiki/Realcompact_space?oldid=625743690 Contributors: D6, Vipul, R.e.b., Sillyrabbit, CBM, Myasuda, Jac16888, Cydebot, Headbomb, Agricola44, Jaan Vajakas, Addbot, Erik9bot, Marcus0107, Schojoha, Markviking, Chen10k2 and Anonymous: 3

• Regular space Source: http://en.wikipedia.org/wiki/Regular_space?oldid=611251026 Contributors: AxelBoldt, Toby Bartels, Patrick,Michael Hardy, Charles Matthews, Fibonacci, Tobias Bergemann, Tosha, Giftlite, Fropuff, Waltpohl, Vipul, Margosbot~enwiki, HairyDude, Sardanaphalus, SmackBot, MalafayaBot, Germandemat, Jim.belk, Dp462090, Equendil, Cydebot, Mathematrucker, MetsBot,David Eppstein, Adavidb, Caboose908, PMajer, Ylebru, Arcfrk, AlleborgoBot, Ocsenave, Thehotelambush, MystBot, Addbot, Laaknor-Bot, Peter Grabs, Tohuwaboho, GrouchoBot, Rickhev1, EmausBot, Slawekb, Bezik, Chester Markel, Brad7777 and Anonymous: 9

• Relatively compact subspace Source: http://en.wikipedia.org/wiki/Relatively_compact_subspace?oldid=543779380 Contributors: TobyBartels, Michael Hardy, Revolver, Charles Matthews, ElBenevolente, Tosha, Aylex~enwiki, Linas, Algebraist, Zwobot, SmackBot, Sillyrabbit, DHN-bot~enwiki, Konradek, Sullivan.t.j, Kakila, Addbot, Erik9bot, EmausBot, ZéroBot, CocuBot, PatrickR2 and Anonymous:7

• Second-countable space Source: http://en.wikipedia.org/wiki/Second-countable_space?oldid=624610634Contributors: Michael Hardy,Ciphergoth, Charles Matthews, Fropuff, Paul August, Vipul, Teorth, Crust, Burn, Oleg Alexandrov, Linas, BD2412, R.e.b., Chobot, NolAders, Melchoir, Lambiam, Freelance Intellectual, Cydebot, Eleuther, Myrkkyhammas, RowellSK, UKoch, Addbot, Topology Expert,WuBot, Ptbotgourou, Xqbot, RibotBOT, ZéroBot, Zdorovo, Levap6, Snotbot, MyWikiNik, Minsbot and Anonymous: 9

• Sequence Source: http://en.wikipedia.org/wiki/Sequence?oldid=665140592 Contributors: AxelBoldt, Mav, Zundark, Tarquin, XJaM,Toby Bartels, Imran, Camembert, Youandme, Lir, Patrick, Michael Hardy, Ihcoyc, Poor Yorick, Nikai, EdH, Charles Matthews, Dys-prosia, Greenrd, Hyacinth, Zero0000, Sabbut, Garo, Robbot, Lowellian, MathMartin, Stewartadcock, Henrygb, Bkell, Tosha, Centrx,Giftlite, BenFrantzDale, Lupin, Herbee, Horatio, Edcolins, Vadmium, Leonard Vertighel, Manuel Anastácio, Alexf, Fudo, Melikamp,Sam Hocevar, Tsemii, Ross bencina, Jiy, TedPavlic, Paul August, JoeSmack, Elwikipedista~enwiki, Syp, Pjrich, Shanes, Jonathan Drain,Nk, Obradovic Goran, Haham hanuka, Zaraki~enwiki, Merope, Jumbuck, Reubot, Jet57, Olegalexandrov, Ringbang, Djsasso, Total-cynic, Oleg Alexandrov, Hoziron, Linas, Madmardigan53, MFH, Isnow, Graham87, Dpv, Mendaliv, Salix alba, Figs, VKokielov, Log-gie, Rsenington, RexNL, Pexatus, Fresheneesz, Kri, Ryvr, Chobot, Lightsup55, Krishnavedala, Wavelength, Michael Slone, Grubber,Arthur Rubin, JahJah, Pred, Finell, KHenriksson, Gelingvistoj, Chris the speller, Bluebot, Nbarth, Mcaruso, Suicidalhamster, SundarBot,Dreadstar, Fagstein, Just plain Bill, Xionbox, Dreftymac, Gco, CRGreathouse, CBM, Gregbard, Cydebot, Xantharius, Epbr123, KCliffer,Saber Cherry, Rlupsa, Marek69, Urdutext, Icep, Ste4k, Mutt Lunker, JAnDbot, Asnac, Coolhandscot, Martinkunev, VoABot II, Avjoska,JamesBWatson, Brusegadi, Minimiscience, Stdazi, DerHexer, J.delanoy, Trusilver, Suenm~enwiki, Ncmvocalist, Belovedfreak, Policron,JingaJenga, VolkovBot, ABF, AlnoktaBOT, Philip Trueman, Digby Tantrum, JhsBot, Isis4563, Wolfrock, Xiong Yingfei, Newbyguesses,SieBot, Scarian, Yintan, Xelgen, Outs, Paolo.dL, OKBot, Pagen HD,Wahrmund, Classicalecon, Atif.t2, Crambo0349, ClueBot, Justin WSmith, Fyyer, SuperHamster, Excirial, Estirabot, Jotterbot, Thingg, Downgrader, Aj00200, XLinkBot, Stickee, Rror, WikHead, Brent-smith101, Addbot, Non-dropframe, Kongr43gpen, Matěj Grabovský, Legobot, Luckas-bot, Yobot, Eric-Wester, 4th-otaku, AnomieBOT,

258 CHAPTER 58. Σ-COMPACT SPACE

Jim1138, Law, Materialscientist, E2eamon, ArthurBot, Ayda D, Xqbot, Omnipaedista, RibotBOT, Charvest, Shadowjams, Thehelpful-bot, Dan6hell66, Constructive editor, Mark Renier, Tal physdancer, SixPurpleFish, Pinethicket, BRUTE, SkyMachine, PiRSquared17,Roy McCoy, RjwilmsiBot, Tzfyr, EmausBot, John of Reading, GoingBatty, Wikipelli, K6ka, Brent Perreault, Nellandmice, Bethnim,Ida Shaw, Alpha Quadrant, KuduIO, D.Lazard, SporkBot, Wayne Slam, Donner60, Chewings72, ClueBot NG, Satellizer, Widr, MerlI-wBot, Helpful Pixie Bot, HMSSolent, Curb Chain, Calabe1992, Brad7777, Minsbot, Praxiphenes, EuroCarGT, Ven Seyranyan., Jegyao,DavyRalph, Graphium, Jochen Burghardt, Brirush, Mark viking, LoMaPh, Immonster, EricsonWillians, Emlynlee, Buscus 3, JackHoang,Some1Redirects4You and Anonymous: 209

• Sequentially compact space Source: http://en.wikipedia.org/wiki/Sequentially_compact_space?oldid=656966423Contributors: MichaelShulman, TakuyaMurata, Tobias Bergemann, Tosha, Smjg, Fropuff, Paul August, StradivariusTV, Algebraist, RonnieBrown, Silly rab-bit, Obueno~enwiki, Jay Gatsby, Plclark, Talan Gwynek, Addbot, Topology Expert, Citation bot, Edsegal, J04n, Point-set topologist,FrescoBot, EmausBot, WikitanvirBot, ZéroBot, Chharvey, Helpful Pixie Bot, Paolo Lipparini, Mark viking and Anonymous: 13

• Set (mathematics) Source: http://en.wikipedia.org/wiki/Set_(mathematics)?oldid=667480360 Contributors: Damian Yerrick, Axel-Boldt, Lee Daniel Crocker, Archibald Fitzchesterfield, Uriyan, Bryan Derksen, Zundark, The Anome, -- April, LA2, XJaM, Toby~enwiki,Toby Bartels, Pb~enwiki, Waveguy, LapoLuchini, Youandme, Olivier, Paul Ebermann, Frecklefoot, Patrick, TeunSpaans, Michael Hardy,Wshun, Booyabazooka, Fred Bauder, Nixdorf, Wapcaplet, TakuyaMurata, CesarB, Mkweise, Iulianu, Docu, Den fjättrade ankan~enwiki,Александър, UserGoogol, Rob Hooft, Jonik, Mxn, Etaoin, Vargenau, Pizza Puzzle, Schneelocke, Ideyal, Charles Matthews, Timwi,Dcoetzee, Ike9898, Dysprosia, Jitse Niesen, Greenrd, Prumpf, Furrykef, Hyacinth, Saltine, Stormie, RealLink, JorgeGG, Phil Boswell,Robbot, Astronautics~enwiki, Fredrik, Altenmann, Peak, Romanm, Mfc, Tobias Bergemann, Centrx, Giftlite, Smjg, Ævar ArnfjörðBjarmason, Lethe, Dissident, Waltpohl, Ezhiki, Prosfilaes, Python eggs, Chameleon, Gubbubu, Leonard Vertighel, Utcursch, Gdr, Knu-tux, Antandrus, BozMo, Mustafaa, Rousearts, Joseph Myers, Crawdaddio, Maximaximax, Wiml, Tothebarricades.tk, Zfr, Sam Hoce-var, Urhixidur, Vivacissamamente, Porges, Corti, PhotoBox, Shahab, Brianjd, Dissipate, Vinoir, EugeneZelenko, Discospinster, Mani1,Paul August, Demaag, Rgdboer, Crisófilax, Aaronbrick, Bobo192, Army1987, C S, Johnteslade, Blotwell, ריינהארט ,לערי Jumbuck,Msh210, Gary, Tablizer, Eric Kvaalen, Andrewpmk, Richard Fannin, EvenT, Arag0rn, CloudNine, Dirac1933, Spambit, Oleg Alexan-drov, Kendrick Hang, Scndlbrkrttmc, Ott, OwenX, Prashanthns, Marudubshinki, LimoWreck, Graham87, BD2412, Dpr, Josh Par-ris, Mayumashu, MarSch, Trlovejoy, Salix alba, Heah, Bubba73, VKokielov, Mathbot, Crazycomputers, Jrtayloriv, Fresheneesz, Kri,Chobot, Algebraist, YurikBot, Wavelength, Xcelerate, Charles Gaudette, RussBot, Lucinos~enwiki, Thane, Trovatore, Srinivasasha,BOT-Superzerocool, Bota47, RyanJones, Ms2ger, Hirak 99, Lt-wiki-bot, Arthur Rubin, Gulliveig, Reyk, ArielGold, Finell, Dudzcom,RupertMillard, SmackBot, Unyoyega, Bomac, Brick Thrower, BiT, Gilliam, Kaiserb, @modi, Trebor, MartinPoulter, Jerome CharlesPotts, Octahedron80, DHN-bot~enwiki, Bob K, Cybercobra, Nakon, Jiddisch~enwiki, Richard001, MathStatWoman, Just plain Bill,RayGates, SashatoBot, Dfass, Loadmaster, Squigglet, Rosejn, IvanLanin, Tawkerbot2, JRSpriggs, KNM, CRGreathouse, Benjistern,Ale jrb, Makeemlighter, CBM, Anakata, Except, Sax Russell, ShelfSkewed, WeggeBot, Asztal, Gregbard, Danman3459, MKil, Tick-etMan, Julian Mendez, He Who Is, Viridae, Xantharius, Lindsay658, Daniel Olsen, Malleus Fatuorum, Thijs!bot, Epbr123, Knakts,RobHar, Escarbot, EJR~enwiki, Seaphoto, Tchakra, JAnDbot, Leuko, Tomst, VoABot II, JNW, Kajasudhakarababu, Echoback, DavidEppstein, JoergenB, Sammi84, MartinBot, Vigyani, J.delanoy, Maurice Carbonaro, Cpiral, Utkwes, NewEnglandYankee, DavidCBryant,Djr13, Idioma-bot, VolkovBot, Paul.w.bennett, Am Fiosaigear~enwiki, Philip Trueman, TXiKiBoT, Anonymous Dissident, Ocolon, Jhs-Bot, Wowzavan, PaulTanenbaum, LBehounek, Lerdthenerd, Synthebot, A Raider Like Indiana, Dmcq, Symane, EmxBot, PaddyLeahy,YohanN7, Dogah, SieBot, BotMultichill, Jauerback, Yintan, Happysailor, Paolo.dL, Allmightyduck, Oxymoron83, Kumioko (renamed),Cyfal, DEMcAdams, WikiBotas, Damien Karras, ClueBot, Arcsecant, PipepBot, Cliff, DionysosProteus, Unbuttered Parsnip, Gogamoga,Drmies, Asdasdasda, Liempt, Jusdafax, Watchduck, Jotterbot, Hans Adler, Apparition11, Gerhardvalentin, Mosaick~enwiki, Ahsanlassi,Multipundit, Addbot, Cxz111, Mnmazur, Protonk, The world deserves the truth, Ingeniosus, Omnipedian, LinkFA-Bot, Harsha6raju,Tide rolls, Lightbot, Zorrobot, Jarble, Yobot, Fraggle81, TonyFlury, AnomieBOT, Materialscientist, Kimsey0, Twri, ArthurBot, Xqbot,Taffer9, Vxk08u, Medoshalaby, Pmlineditor, GrouchoBot, RibotBOT, Mathonius, Fangncurl, Laelele, SD5, FrescoBot, Sławomir Biały,Mfwitten, DivineAlpha, Tkuvho, Pinethicket, NearSetAccount, 10metreh, SkyMachine, Dashed, Tgv8925, Declan Clam, Miracle Pen,Bluefist, Theo10011, Jesse V., DARTH SIDIOUS 2, EmausBot, Acather96, RenamedUser01302013, John Cline, Bollyjeff, Akerans,Hgetnet, L Kensington, Zephyrus Tavvier, Donner60, Chewings72, Puffin, Gwestheimer, Petrb, ClueBot NG, Gareth Griffith-Jones, Satel-lizer, SusikMkr, Frietjes, O.Koslowski, Widr, WikiPuppies, MerlIwBot, Helpful Pixie Bot, BG19bot, Saulpila2000, Shashank rathore,Frze, AvocatoBot, AmieKim, Shyamli rao, Cliff12345, Eduardofeld, Krothgar, Sivarama.prsd, Ejazahmed007, MadGuy7023, Volvens,Mark L MacDonald, Stephan Kulla, Jfanderson68, Epicgenius, Jodosma, Tentinator, Glen Behrend, Bg9989, DavidLeighEllis, Bvraama-raaju, Ugog Nizdast, Ginsuloft, Quenhitran, AddWittyNameHere, Wikireadya, Mahusha, IvanZhilin, Degenerate prodigy, KasparBot, RJraghava, Mohammad Saad Ifrahim Khan and Anonymous: 434

• Strictly singular operator Source: http://en.wikipedia.org/wiki/Strictly_singular_operator?oldid=663253915Contributors: Michael Hardy,Linas, Jmath666, Addbot, AnomieBOT, John of Reading, Mgkrupa and Anonymous: 4

• Subset Source: http://en.wikipedia.org/wiki/Subset?oldid=658760669 Contributors: Damian Yerrick, AxelBoldt, Youssefsan, XJaM,Toby Bartels, StefanRybo~enwiki, Edward, Patrick, TeunSpaans, Michael Hardy, Wshun, Booyabazooka, Ellywa, Oddegg, Andres,Charles Matthews, Timwi, Hyacinth, Finlay McWalter, Robbot, Romanm, Bkell, 75th Trombone, Tobias Bergemann, Tosha, Giftlite,Fropuff, Waltpohl, Macrakis, Tyler McHenry, SatyrEyes, Rgrg, Vivacissamamente, Mormegil, EugeneZelenko, Noisy, Deh, Paul Au-gust, Engmark, Spoon!, SpeedyGonsales, Obradovic Goran, Nsaa, Jumbuck, Raboof, ABCD, Sligocki, Mac Davis, Aquae, LFaraone,Chamaeleon, Firsfron, Isnow, Salix alba, VKokielov, Mathbot, Harmil, BMF81, Chobot, Roboto de Ajvol, YurikBot, Alpt, Dmharvey,KSmrq, NawlinWiki, Trovatore, Nick, Szhaider, Wasseralm, Sardanaphalus, Jacek Kendysz, BiT, Gilliam, Buck Mulligan, SMP, Or-angeDog, Bob K, Dreadstar, Bjankuloski06en~enwiki, Loadmaster, Vedexent, Amitch, Madmath789, Newone, CBM, Jokes Free4Me,345Kai, SuperMidget, Gregbard, WillowW, MC10, Thijs!bot, Headbomb, Marek69, RobHar, WikiSlasher, Salgueiro~enwiki, JAnDbot,.anacondabot, Pixel ;-), Pawl Kennedy, Emw, ANONYMOUS COWARD0xC0DE, RaitisMath, JCraw, Tgeairn, Ttwo, Maurice Car-bonaro, Acalamari, Gombang, NewEnglandYankee, Liatd41, VolkovBot, CSumit, Deleet, Rei-bot, AnonymousDissident, James.Spudeman,PaulTanenbaum, InformationSpace, Falcon8765, AlleborgoBot, P3d4nt, NHRHS2010, Garde, Paolo.dL, OKBot, Brennie8, Jons63,Loren.wilton, ClueBot, GorillaWarfare, PipepBot, The Thing That Should Not Be, DragonBot, Watchduck, Hans Adler, Computer97,Noosentaal, Versus22, PCHS-NJROTC, Andrew.Flock, Reverb123, Addbot, , Fyrael, PranksterTurtle, Numbo3-bot, Zorrobot, Jar-ble, JakobVoss, Luckas-bot, Yobot, Synchronism, AnomieBOT, Jim1138, Materialscientist, Citation bot, Martnym, NFD9001, Char-vest, 78.26, XQYZ, Egmontbot, Rapsar, HRoestBot, Suffusion of Yellow, Agent Smith (The Matrix), RenamedUser01302013, ZéroBot,Alexey.kudinkin, Chharvey, Quondum, Chewings72, 28bot, ClueBot NG, Wcherowi, Matthiaspaul, Bethre, Mesoderm, O.Koslowski,AwamerT, Minsbot, Pratyya Ghosh, YFdyh-bot, Ldfleur, ChalkboardCowboy, Saehry, Stephan Kulla, , Ilya23Ezhov, Sandshark23,Quenhitran, Neemasri, Prince Gull, Maranuel123, Alterseemann, Rahulmr.17 and Anonymous: 179

58.5. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 259

• Subspace topology Source: http://en.wikipedia.org/wiki/Subspace_topology?oldid=667550881 Contributors: Toby Bartels, Karada, Re-volver, MathMartin, Giftlite, Fropuff, Jason Quinn, DemonThing, Thufir Hawat, Paul August, Eric Kvaalen, Linas, Ryan Reich, BD2412,Chobot, YurikBot, Zwobot, Zvika, SmackBot, Mets501,WeggeBot, Kilva, Bwhack, Jwuthe2, LokiClock, Arcfrk, VVVBot, JackSchmidt,Alexbot, Jaan Vajakas, Addbot, Yobot, Calle, KamikazeBot, LilHelpa, Xqbot, Trappist the monk, EmausBot, WikitanvirBot and Anony-mous: 17

• Supercompact space Source: http://en.wikipedia.org/wiki/Supercompact_space?oldid=655715022Contributors: Zundark,Michael Hardy,Rich Farmbrough, Vipul, C S, RJFJR, OdedSchramm, Rjwilmsi, SmackBot, Silly rabbit, Baa, TenPoundHammer, Lambiam, Olaf Davis,Cydebot, David Eppstein, TallNapoleon, Anturiaethwr, Topology Expert, Citation bot, Citation bot 1, Trappist the monk, Suslindisam-biguator, Helpful Pixie Bot and Anonymous: 1

• Topological space Source: http://en.wikipedia.org/wiki/Topological_space?oldid=662870663 Contributors: AxelBoldt, Zundark, XJaM,Toby Bartels, Olivier, Patrick, Michael Hardy, Wshun, Kku, Dineshjk, Karada, Hashar, Zhaoway~enwiki, Revolver, Charles Matthews,Dcoetzee, Dysprosia, Kbk, Taxman, Phys, Robbot, Nizmogtr, Fredrik, Saaska, MathMartin, P0lyglut, Tobias Bergemann, Giftlite, GeneWard Smith, Lethe, Fropuff, Dratman, DefLog~enwiki, Rhobite, Luqui, Paul August, Dolda2000, Elwikipedista~enwiki, Tompw, Aude,SgtThroat, Tsirel, Marc van Woerkom, Varuna, Kuratowski’s Ghost, Msh210, Keenan Pepper, Danog, Sligocki, Spambit, Oleg Alexan-drov, Woohookitty, Graham87, BD2412, Grammarbot, FlaBot, Sunayana, Tillmo, Chobot, Algebraist, YurikBot, Wavelength, HairyDude, NawlinWiki, Rick Norwood, Bota47, Stefan Udrea, Hirak 99, Arthur Rubin, Lendu, JoanneB, Eaefremov, RonnieBrown, Sar-danaphalus, SmackBot, Maksim-e~enwiki, Sciyoshi~enwiki, DHN-bot~enwiki, Tsca.bot, Tschwenn, LkNsngth, Vriullop, Arialblack, Iri-descent, Devourer09,Mattbr, AndrewDelong, Kupirijo, Roccorossi, Xantharius, Thijs!bot, Konradek, Odoncaoa, Escarbot, Salgueiro~enwiki,JAnDbot, YK Times, Jakob.scholbach, Bbi5291, Wdevauld, J.delanoy, Pharaoh of the Wizards, Maurice Carbonaro, The Mudge, Jma-jeremy, Policron, TXiKiBoT,AnonymousDissident, Plclark, AaronRotenberg, Jesin, Arcfrk, SieBot, MiNombreDeGuerra, JerroldPease-Atlanta, JackSchmidt, Failure.exe, Egmontaz, Palnot, SilvonenBot, Addbot, CarsracBot, AnnaFrance, ChenzwBot, Luckas-bot, Yobot,SwisterTwister, AnomieBOT, Ciphers, Materialscientist, Citation bot, DannyAsher, FlordiaSunshine342, J04n, Point-set topologist, Fres-coBot, Jschnur, Jeroen De Dauw, TobeBot, Seahorseruler, Skakkle, Cstanford.math, ZéroBot, Chharvey, Wikfr, Orange Suede Sofa,Liuthar, ClueBot NG, Wcherowi, Mesoderm, Vinícius Machado Vogt, Helpful Pixie Bot, Gaurav Nirala, Tom.hirschowitz, Pacerier,Cpatra1984, Brad7777, Minsbot, LoganFromSA, MikeHaskel, Acer4666, Freeze S, Mark viking, Epicgenius, Kurt Artindagi, Improba-ble keeler, Amonk1962, KasparBot and Anonymous: 108

• Topology Source: http://en.wikipedia.org/wiki/Topology?oldid=667175062 Contributors: AxelBoldt, Brion VIBBER, Zundark, TheAnome, XJaM, Vanderesch, Toby~enwiki, Toby Bartels, Hhanke, Miguel~enwiki, Camembert, Hephaestos, Olivier, Bdesham, Patrick,Michael Hardy, Wshun, Liftarn, Gnomon~enwiki, Sannse, TakuyaMurata, GTBacchus, Karada, MightCould, CesarB, Cyp, Mark Fos-key, Mxn, Darkov, Charles Matthews, Dcoetzee, Dino, Sicro, Dysprosia, Jitse Niesen, Wik, Zoicon5, Steinsky, Hyacinth, Saltine,Jeanmichel~enwiki, Jusjih, Robbot, Mountain, Donreed, Altenmann, George Kontopoulos, Gandalf61, MathMartin, Henrygb, Rursus,Robinh, Fuelbottle, TexasDex, Tobias Bergemann, Ramir, Pdenapo, Weialawaga~enwiki, Marc Venot, Tosha, Kevin Saff, Giftlite, Dben-benn, Jyril, Rudolf 1922, Inter, Lethe, Fropuff, Everyking, Curps, Guanaco, Jason Quinn, Ptk~enwiki, Prosfilaes, Dan Gardner, Gadfium,LucasVB, Onco p53, APH, Maximaximax, Gauss, Abdull, ELApro, D6, Ta bu shi da yu, Imroy, Discospinster, Rich Farmbrough, Lind-sayH, Paul August, Dmr2, Violetriga, Gauge, Tompw, El C, Rgdboer, Aude, EmilJ, Keron Cyst, C S, Shenme, Jjk, Jung dalglish, Maur-reen, Haham hanuka, Mdd, Varuna, Jumbuck,Msh210, Danog, Sligocki, Derbeth, OlegAlexandrov, Brookie, Saeed, Velho,Woohookitty,Linas, Spamguy, Prophile, Oliphaunt, WadeSimMiser, Orz, MONGO, Dzordzm, Graham87, Magister Mathematicae, Porcher, Rjwilmsi,Mayumashu, Tangotango, Hychu, Salix alba, NonNobis~enwiki, Yamamoto Ichiro, FlaBot, Nivix, Isotope23, Windharp, Chobot, DylanThurston, Gdrbot, Algebraist, Wavelength, Borgx, Hairy Dude, Hede2000, Stephenb, Chaos, Cryptic, Rick Norwood, Joth, The Ogre,Trovatore, TechnoGuyRob, Crasshopper, Natkeeran, Aaron Schulz, EEMIV, Aidanb, PyroGamer, User27091, Stefan Udrea, Tetracube,Arthur Rubin, Bentong Isles, Naught101, Curpsbot-unicodify, Ilmari Karonen, TMott, RonnieBrown, Brentt, Sardanaphalus, SmackBot,RDBury, Mmernex, David Kernow, Honza Záruba, KnowledgeOfSelf, Delldot, Cokebingo, Alsandro, Wikikris, Gilliam, Betacommand,Skizzik, Chaojoker, Bluebot, The baron, MalafayaBot, Stevage, Darth Panda, Nick Levine, Alriode, Mhym, Lesnail, Jackohare, Ran-domP, Geoffr, Jon Awbrey, Sammy1339, Dr. Gabriel Gojon, Ohconfucius, Lambiam, Rory096, NongBot~enwiki, Atoll, Mr Stephen,Thevelho, Stephen B Streater, Jason.grossman, Madmath789, Francl, Jbolden1517, CRGreathouse, CmdrObot, Jrolland, CBM, Ranicki,Usgnus, Werratal, Myasuda, Yaris678, Cydebot, Gogo Dodo, Corpx, Dr.enh, Starship Trooper, Gaoos, Mariontte User, Dharma6662000,Thijs!bot, Epbr123, J. Charles Taylor, Knakts, Perrygogas, JustAGal, Chadnash, AbcXyz, Escarbot, LachlanA, AntiVandalBot, Seaphoto,Tchakra, Emeraldcityserendipity, Weixifan, Lfstevens, Byrgenwulf, Ioeth, GromXXVII, Turgidson, MER-C, Skomorokh, Magioladitis,Jéské Couriano, SwiftBot, Bubba hotep, Ensign beedrill, Sullivan.t.j, SnakeChess5, Pax:Vobiscum, Rootneg2, MartinBot, Erkan Yil-maz, J.delanoy, Maurice Carbonaro, MarcoLittel, Policron, Jamesofur, Alan U. Kennington, Jeff G., JohnBlackburne, Nousernamesleft,Topologyxpert, TXiKiBoT, Ttopperr, Philosotox, Anonymous Dissident, Qxz, Digby Tantrum, Softkitten88, ARUNKUMAR P.R, Wol-frock, Enviroboy, Dmcq, MiamiMath, Symane, ADOGisAboat, Katzmik, Rknasc, Rybu, YohanN7, SieBot, Ivan Štambuk, Meldor, Tri-wbe, Aristolaos, Nicinic, Pendlehaven, Daniarmo, MiNombreDeGuerra, Jorgen W, Kumioko, Valeria.depaiva, Vituzzu, Laurentseries,Jludwig, ClueBot, The Thing That Should Not Be, Stevanspringer, TheSmuel, Monty42, SapphireJay, SchreiberBike, Triathematician,Manatee331, Robertabrams, Novjunulo, Fastily, Pi.C.Noizecehx, ErickOrtiz, Tilmanbauer, MystBot, Addbot, Some jerk on the Internet,Yobmod, Fieldday-sunday, MrOllie, CarsracBot, Bazza1971, LinkFA-Bot, Jasper Deng, K-topology, Tide rolls, Lightbot, OlEnglish,Zorrobot, TeH nOmInAtOr, Jarble, Sammtamm, Legobot, Cote d'Azur, Luckas-bot, 2D, Deputyduck, AnomieBOT, 1exec1, Jack-ieBot, AdjustShift, Materialscientist, Citation bot, Xqbot, Ekwos, J04n, GrouchoBot, Point-set topologist, RibotBOT, Charvest, Con-traverse, Divisbyzero, Orhanghazi, VI, Anilkumarphysics, Commit charge, Pinethicket, Tom.Reding, CrowzRSA, PoincaresChild, To-beBot, Jws401, Enthdegree, Integrals4life, Unbitwise, Jesse V., EmausBot, Fly by Night, Slawekb, ZéroBot, Chimpdmunk, The Nut,Caspertheghost, QEDK, Staszek Lem, Lorem Ip, ProteoPhenom, Anita5192, ResearchRave, ClueBot NG, Wcherowi, O.Koslowski,Widr, Helpful Pixie Bot, Daheadhunter, BG19bot, TCN7JM, Bigdon128, Wimvdam, Brad7777, Charismaa, Waleed.598, Sboosali, JY-Bot, MrBubbleFace, Dexbot, Paulo Henrique Macedo, King jakob c, Brirush, Mark viking, Ayesh2788, I am One of Many, TJLaher123,SakeUPenn, K401sTL3, Lizia7, Sesamo12, Btomoiaga, Chuluojun, Je.est.un.autre, Betapictoris, SoSivr, Jainmskip, Hriton, KasparBotand Anonymous: 348

• Total order Source: http://en.wikipedia.org/wiki/Total_order?oldid=666410207 Contributors: Damian Yerrick, AxelBoldt, Zundark,XJaM, Fritzlein, Patrick, Michael Hardy, Dori, AugPi, Dysprosia, Jitse Niesen, Greenrd, Zoicon5, Hyacinth, VeryVerily, Fibonacci,McKay, Aleph4, Gandalf61, MathMartin, Rursus, Tobias Bergemann, Giftlite, Mshonle~enwiki, Markus Krötzsch, Lethe, Waltpohl,DefLog~enwiki, Alberto da Calvairate~enwiki, Quarl, Elroch, Paul August, Susvolans, Army1987, Func, Cmdrjameson, Msh210, Pion,Joriki, MattGiuca, Yurik, OneWeirdDude, Salix alba, VKokielov, Mathbot, Margosbot~enwiki, Wastingmytime, Chobot, YurikBot,Hede2000, Tetracube, Rdore, Melchoir, Gelingvistoj, Mhss, Chris the speller, Bazonka, Jdthood, Javalenok, Michael Kinyon, Loadmas-ter, Mets501, George100, CRGreathouse, CBM, Thomasmeeks, Oryanw~enwiki, VectorPosse, JAnDbot, David Eppstein, Infovarius,

260 CHAPTER 58. Σ-COMPACT SPACE

Osquar F, PaulTanenbaum, SieBot, Ceroklis, Anchor Link Bot, Heinzi.at, WurmWoode, Universityuser, Palnot, Marc van Leeuwen,Addbot, Tanhabot, AsphyxiateDrake, Luckas-bot, Yobot, Charlatino, White gecko, 1exec1, Infvwl, GrouchoBot, Jsjunkie, Quondum,D.Lazard, SporkBot, CocuBot, BG19bot, YumOooze, YFdyh-bot, Austinfeller, Mark viking, नितीश चनदर and Anonymous: 47

• Totally bounded space Source: http://en.wikipedia.org/wiki/Totally_bounded_space?oldid=652130953 Contributors: Awaterl, TobyBartels, MathMartin, Tobias Bergemann, Fropuff, Paul August, Oleg Alexandrov, Simetrical, Koroner, Zwobot, Kompik, Silly rabbit,A Geek Tragedy, Dreadstar, JHunterJ, Thijs!bot, Salgueiro~enwiki, GromXXVII, McM.bot, Megaloxantha, M.landgren, Niceguyedc,Luca Antonelli, Addbot, Luckas-bot, Yobot, Citation bot, Lost-n-translation, DrilBot, Junior Wrangler, Espresso-hound, WikitanvirBot,DPL bot, Noix07 and Anonymous: 24

• Tychonoff’s theorem Source: http://en.wikipedia.org/wiki/Tychonoff’s_theorem?oldid=630581362 Contributors: AxelBoldt, Zun-dark, Toby Bartels, Michael Hardy, Chinju, TakuyaMurata, Poor Yorick, Revolver, Charles Matthews, Dcoetzee, Dysprosia, Wjcook,Fibonacci, ,דוד MathMartin, Choni, Tobias Bergemann, Giftlite, Markus Krötzsch, Lupin, Bender235, Momotaro, Haham hanuka, Jin-hyun park, Arthena, ABCD, Oleg Alexandrov, Linas, Ruud Koot, Ryan Reich, Mathbot, Sodin, Chobot, Hairy Dude, Schmock, Hen-nobrandsma, Raijinili, Zvika, AndrewWTaylor, RDBury, BeteNoir, Incnis Mrsi, Autarch, Shalom Yechiel, Yosha, Grasyop, Mets501,MarkusQ, Jasperdoomen, Amit Moscovich, Headbomb, JAnDbot, Albmont, Sullivan.t.j, David Eppstein, PMajer, Plclark, Selbiniyaz,Dogah, SieBot, ToePeu.bot, Soler97, Niceguyedc, Alexey Muranov, Ncsinger, Addbot, SpBot, Legobot, Luckas-bot, Yobot, Point-settopologist, Louperibot, Trappist the monk, Xnn, EmausBot, ZéroBot, Helpful Pixie Bot, Brad7777, Ant314159265, Reader634 andAnonymous: 29

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262 CHAPTER 58. Σ-COMPACT SPACE

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264 CHAPTER 58. Σ-COMPACT SPACE

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