Locally Normal Space 4

Locally normal space 4Wikipedia

Contents

1 Alexandrov topology 11.1 Characterizations of Alexandrov topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Duality with preordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 The Alexandrov topology on a preordered set . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 The specialization preorder on a topological space . . . . . . . . . . . . . . . . . . . . . . 21.2.3 Equivalence between preorders and Alexandrov topologies . . . . . . . . . . . . . . . . . 21.2.4 Equivalence between monotony and continuity . . . . . . . . . . . . . . . . . . . . . . . . 31.2.5 Category theoretic description of the duality . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.6 Relationship to the construction of modal algebras from modal frames . . . . . . . . . . . 4

1.3 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Approach space 62.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Equivalent denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Categorical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Baire space 93.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2.1 Modern denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2.2 Historical denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.4 Baire category theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.5 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.8 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

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4 Baire space (set theory) 124.1 Topology and trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.3 Relation to the real line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 Base (topology) 145.1 Simple properties of bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.2 Objects dened in terms of bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.3 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.4 Base for the closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.5 Weight and character . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5.5.1 Increasing chains of open sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

6 Borel set 186.1 Generating the Borel algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

6.1.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.2 Standard Borel spaces and Kuratowski theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.3 Non-Borel sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.4 Alternative non-equivalent denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

7 Boundary (topology) 227.1 Common denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247.4 Boundary of a boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

8 Bounded set 268.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.2 Metric space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.3 Boundedness in topological vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.4 Boundedness in order theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

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9 Category (mathematics) 299.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309.3 Small and large categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319.5 Construction of new categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

9.5.1 Dual category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319.5.2 Product categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

9.6 Types of morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329.7 Types of categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

10 Category of topological spaces 3510.1 As a concrete category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3510.2 Limits and colimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3610.3 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3610.4 Relationships to other categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3610.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

11 Category theory 3811.1 An abstraction of other mathematical concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3911.2 Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

11.2.1 Categories, objects, and morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3911.2.2 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4011.2.3 Natural transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

11.3 Categories, objects, and morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4011.3.1 Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4011.3.2 Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

11.4 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4111.5 Natural transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4211.6 Other concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

11.6.1 Universal constructions, limits, and colimits . . . . . . . . . . . . . . . . . . . . . . . . . 4211.6.2 Equivalent categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4311.6.3 Further concepts and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4311.6.4 Higher-dimensional categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

11.7 Historical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4411.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4411.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4511.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4511.11Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

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11.12External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

12 Cauchy sequence 4812.1 In real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4912.2 In a metric space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4912.3 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

12.3.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4912.3.2 Counter-example: rational numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4912.3.3 Counter-example: open interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5012.3.4 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

12.4 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5112.4.1 In topological vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5112.4.2 In topological groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5112.4.3 In groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5112.4.4 In constructive mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5112.4.5 In a hyperreal continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

12.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5212.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5212.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5212.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

13 Clopen set 5313.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5413.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5413.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5413.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

14 Closed set 5514.1 Equivalent denitions of a closed set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5514.2 Properties of closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5514.3 Examples of closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5514.4 More about closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5614.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5614.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

15 Closure (topology) 5715.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

15.1.1 Point of closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5715.1.2 Limit point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5715.1.3 Closure of a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

15.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5815.3 Closure operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5915.4 Facts about closures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

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15.5 Categorical interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6015.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6015.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6015.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6015.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

16 Compact space 6116.1 Historical development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6216.2 Basic examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6316.3 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

16.3.1 Open cover denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6316.3.2 Equivalent denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6416.3.3 Compactness of subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

16.4 Properties of compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6516.4.1 Functions and compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6516.4.2 Compact spaces and set operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6516.4.3 Ordered compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

16.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6616.5.1 Algebraic examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

16.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6716.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6816.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6816.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

17 Compact-open topology 7017.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7017.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7017.3 Frchet dierentiable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7117.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7117.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

18 Comparison of topologies 7218.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7218.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7218.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7218.4 Lattice of topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7318.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7318.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7318.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

19 Complement (set theory) 7419.1 Relative complement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7419.2 Absolute complement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

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19.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7619.4 Complements in various programming languages . . . . . . . . . . . . . . . . . . . . . . . . . . . 7619.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7819.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7819.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

20 Complete metric space 7920.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7920.2 Some theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8020.3 Completion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8020.4 Topologically complete spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8120.5 Alternatives and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8120.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8120.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8220.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

21 Connected space 8321.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

21.1.1 Connected components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8421.1.2 Disconnected spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

21.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8421.3 Path connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8521.4 Arc connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8621.5 Local connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8621.6 Set operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8621.7 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8921.8 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8921.9 Stronger forms of connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9021.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9021.11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

21.11.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9021.11.2 General references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

22 Consistency 9122.1 Consistency and completeness in arithmetic and set theory . . . . . . . . . . . . . . . . . . . . . . 9122.2 First-order logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

22.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9222.2.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9222.2.3 Basic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9222.2.4 Henkins theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9322.2.5 Sketch of proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

22.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

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22.4 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9322.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9422.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

23 Continuous function 9523.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9523.2 Real-valued continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

23.2.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9523.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9823.2.3 Non-examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10123.2.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10223.2.5 Directional and semi-continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

23.3 Continuous functions between metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10423.3.1 Uniform, Hlder and Lipschitz continuity . . . . . . . . . . . . . . . . . . . . . . . . . . 105

23.4 Continuous functions between topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 10523.4.1 Alternative denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10723.4.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10823.4.3 Homeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10923.4.4 Dening topologies via continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . 109

23.5 Related notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10923.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11023.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11023.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

24 Contractible space 11224.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11224.2 Locally contractible spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11224.3 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11224.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

25 Cosmic space 11425.1 Examples and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11425.2 Unsolved problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11425.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11425.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

26 Countable set 11526.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11526.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11526.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11526.4 Formal denition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11626.5 Minimal model of set theory is countable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12126.6 Total orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

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26.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12226.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12226.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12226.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

27 Cover (topology) 12327.1 Cover in topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12327.2 Renement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12327.3 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12427.4 Covering dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12427.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12427.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12527.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12527.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

28 Disjoint union 12628.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12628.2 Set theory denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12628.3 Category theory point of view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12728.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12728.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

29 Finite set 12829.1 Denition and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12829.2 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12829.3 Necessary and sucient conditions for niteness . . . . . . . . . . . . . . . . . . . . . . . . . . . 12929.4 Foundational issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13029.5 Set-theoretic denitions of niteness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

29.5.1 Other concepts of niteness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13129.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13129.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13129.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13229.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

30 H-closed space 13330.1 Examples and equivalent formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13330.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13330.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

31 Hausdor space 13431.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13431.2 Equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13531.3 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

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31.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13531.5 Preregularity versus regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13631.6 Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13631.7 Algebra of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13731.8 Academic humour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13731.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13731.10Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13731.11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

32 Homeomorphism 13832.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13832.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

32.2.1 Non-examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13932.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14032.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14032.5 Informal discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14132.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14132.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14132.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

33 Identity function 14233.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14333.2 Algebraic property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14333.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14333.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14333.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

34 If and only if 14434.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14434.2 Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

34.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14434.2.2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14534.2.3 Origin of i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

34.3 Distinction from if and only if . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14534.4 More general usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14634.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14634.6 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14634.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

35 Image 14735.1 Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14735.2 Imagery (literary term) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14735.3 Moving image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

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35.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14835.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14835.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

36 Image (mathematics) 15236.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

36.1.1 Image of an element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15336.1.2 Image of a subset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15336.1.3 Image of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

36.2 Inverse image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15336.3 Notation for image and inverse image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

36.3.1 Arrow notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15336.3.2 Star notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15336.3.3 Other terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

36.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15436.5 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15436.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15536.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15536.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

37 Kolmogorov space 15637.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15637.2 Examples and nonexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

37.2.1 Spaces which are not T0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15637.2.2 Spaces which are T0 but not T1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

37.3 Operating with T0 spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15737.4 The Kolmogorov quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15737.5 Removing T0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15837.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

38 Kuratowski closure axioms 15938.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15938.2 Connection to other axiomatizations of topology . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

38.2.1 Induction of Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15938.2.2 Induction of closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16038.2.3 Recovering notions from topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

38.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16038.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16038.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16038.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

39 Limit point 16139.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

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39.2 Types of limit points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16139.3 Some facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16239.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16239.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

40 Locally compact space 16440.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16440.2 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

40.2.1 Compact Hausdor spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16540.2.2 Locally compact Hausdor spaces that are not compact . . . . . . . . . . . . . . . . . . . 16540.2.3 Hausdor spaces that are not locally compact . . . . . . . . . . . . . . . . . . . . . . . . 16540.2.4 Non-Hausdor examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

40.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16640.3.1 The point at innity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16640.3.2 Locally compact groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

40.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16740.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

41 Locally Hausdor space 16841.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

42 Locally normal space 16942.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16942.2 Examples and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16942.3 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16942.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16942.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

43 Locally regular space 17143.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17143.2 Examples and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17143.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17143.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

44 Mathematical analysis 17244.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17344.2 Important concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

44.2.1 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17444.2.2 Sequences and limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

44.3 Main branches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17544.3.1 Real analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17544.3.2 Complex analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17544.3.3 Functional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

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44.3.4 Dierential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17544.3.5 Measure theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17644.3.6 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

44.4 Other topics in mathematical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17644.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

44.5.1 Physical sciences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17744.5.2 Signal processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17744.5.3 Other areas of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

44.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17744.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17844.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17944.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

45 Meagre set 18045.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

45.1.1 Relation to Borel hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18045.2 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18045.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18145.4 BanachMazur game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18145.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

45.5.1 Subsets of the reals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18145.5.2 Function spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

45.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18145.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18145.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

46 Metric space 18346.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18346.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18346.3 Examples of metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18446.4 Open and closed sets, topology and convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 18546.5 Types of metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

46.5.1 Complete spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18546.5.2 Bounded and totally bounded spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18646.5.3 Compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18746.5.4 Locally compact and proper spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18746.5.5 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18746.5.6 Separable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

46.6 Types of maps between metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18746.6.1 Continuous maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18846.6.2 Uniformly continuous maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18846.6.3 Lipschitz-continuous maps and contractions . . . . . . . . . . . . . . . . . . . . . . . . . 188

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46.6.4 Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18946.6.5 Quasi-isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

46.7 Notions of metric space equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18946.8 Topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18946.9 Distance between points and sets; Hausdor distance and Gromov metric . . . . . . . . . . . . . . 19046.10Product metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

46.10.1 Continuity of distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19046.11Quotient metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19146.12Generalizations of metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

46.12.1 Metric spaces as enriched categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19146.13See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19246.14Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19246.15References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19346.16External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

47 Metrization theorem 19447.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19447.2 Metrization theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19447.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19547.4 Examples of non-metrizable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19547.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19547.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

48 Morphism 19648.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19648.2 Some special morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19748.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19848.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19848.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19848.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19848.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

49 Normal space 19949.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19949.2 Examples of normal spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20049.3 Examples of non-normal spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20049.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20149.5 Relationships to other separation axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20149.6 Citations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20149.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

50 Open and closed maps 20250.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

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50.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20250.3 Open and closed mapping theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20350.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20350.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

51 Partially ordered set 20551.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20651.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20651.3 Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20651.4 Orders on the Cartesian product of partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . 20751.5 Sums of partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20751.6 Strict and non-strict partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20851.7 Inverse and order dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20851.8 Mappings between partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20851.9 Number of partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20951.10Linear extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20951.11In category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21051.12Partial orders in topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21051.13Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21051.14See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21051.15Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21151.16References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21151.17External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

52 Regular space 21252.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21252.2 Relationships to other separation axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21352.3 Examples and nonexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21352.4 Elementary properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21452.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

53 Separated sets 21553.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21553.2 Relation to separation axioms and separated spaces . . . . . . . . . . . . . . . . . . . . . . . . . 21653.3 Relation to connected spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21653.4 Relation to topologically distinguishable points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21653.5 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

54 Separation axiom 21754.1 Preliminary denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21754.2 Main denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21854.3 Relationships between the axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21954.4 Other separation axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

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54.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22054.6 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22054.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

55 Sigma-algebra 22355.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

55.1.1 Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22355.1.2 Limits of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22455.1.3 Sub -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

55.2 Denition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22555.2.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22555.2.2 Dynkins - theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22555.2.3 Combining -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22555.2.4 -algebras for subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22655.2.5 Relation to -ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22655.2.6 Typographic note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

55.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22755.3.1 Simple set-based examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22755.3.2 Stopping time sigma-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

55.4 -algebras generated by families of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22755.4.1 -algebra generated by an arbitrary family . . . . . . . . . . . . . . . . . . . . . . . . . . 22755.4.2 -algebra generated by a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22755.4.3 Borel and Lebesgue -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22855.4.4 Product -algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22855.4.5 -algebra generated by cylinder sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22855.4.6 -algebra generated by random variable or vector . . . . . . . . . . . . . . . . . . . . . . 22955.4.7 -algebra generated by a stochastic process . . . . . . . . . . . . . . . . . . . . . . . . . . 229

55.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22955.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23055.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

56 Subspace topology 23156.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23156.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23156.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23256.4 Preservation of topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23356.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23356.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

57 T1 space 23457.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23457.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

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57.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23557.4 Generalisations to other kinds of spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23657.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

58 Topological space 23758.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

58.1.1 Neighbourhoods denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23758.1.2 Open sets denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23858.1.3 Closed sets denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23958.1.4 Other denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

58.2 Comparison of topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23958.3 Continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23958.4 Examples of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24058.5 Topological constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24158.6 Classication of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24158.7 Topological spaces with algebraic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24158.8 Topological spaces with order structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24158.9 Specializations and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24158.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24258.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24258.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24258.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

59 Tychono space 24459.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24459.2 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24459.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

59.3.1 Preservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24559.3.2 Real-valued continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24559.3.3 Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24659.3.4 Compactications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24659.3.5 Uniform structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

59.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

60 Uniform space 24760.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

60.1.1 Entourage denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24760.1.2 Pseudometrics denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24860.1.3 Uniform cover denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

60.2 Topology of uniform spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24860.2.1 Uniformizable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

60.3 Uniform continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

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60.4 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24960.4.1 Hausdor completion of a uniform space . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

60.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25060.6 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25160.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25160.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

61 Upper set 25261.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25361.2 Ordinal numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25361.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25361.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

62 Vacuous truth 25462.1 Scope of the concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25462.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25462.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25562.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25562.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25562.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25562.7 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 256

62.7.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25662.7.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26462.7.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

Chapter 1

Alexandrov topology

In topology, an Alexandrov space (or Alexandrov-discrete space) is a topological space in which the intersectionof any family of open sets is open. It is an axiom of topology that the intersection of any nite family of open sets isopen. In an Alexandrov space the nite restriction is dropped.Alexandrov topologies are uniquely determined by their specialization preorders. Indeed, given any preorder on aset X, there is a unique Alexandrov topology on X for which the specialization preorder is . The open sets are justthe upper sets with respect to . Thus, Alexandrov topologies on X are in one-to-one correspondence with preorderson X.Alexandrov spaces are also called nitely generated spaces since their topology is uniquely determined by the familyof all nite subspaces. Alexandrov spaces can be viewed as a generalization of nite topological spaces.

1.1 Characterizations of Alexandrov topologiesAlexandrov topologies have numerous characterizations. Let X = be a topological space. Then the followingare equivalent:

Open and closed set characterizations: Open set. An arbitrary intersection of open sets in X is open. Closed set. An arbitrary union of closed sets in X is closed.

Neighbourhood characterizations: Smallest neighbourhood. Every point of X has a smallest neighbourhood. Neighbourhood lter. The neighbourhood lter of every point in X is closed under arbitrary intersec-

tions.

Interior and closure algebraic characterizations: Interior operator. The interior operator of X distributes over arbitrary intersections of subsets. Closure operator. The closure operator of X distributes over arbitrary unions of subsets.

Preorder characterizations: Specialization preorder. T is the nest topology consistent with the specialization preorder of X i.e.

the nest topology giving the preorder satisfying x y if and only if x is in the closure of {y} in X. Open up-set. There is a preorder such that the open sets of X are precisely those that are upwardly

closed i.e. if x is in the set and x y then y is in the set. (This preorder will be precisely the specializationpreorder.)

1

2 CHAPTER 1. ALEXANDROV TOPOLOGY

Closed down-set. There is a preorder such that the closed sets of X are precisely those that aredownwardly closed i.e. if x is in the set and y x then y is in the set. (This preorder will be precisely thespecialization preorder.)

Upward interior. A point x lies in the interior of a subset S of X if and only if there is a point y in Ssuch that y x where is the specialization preorder i.e. y lies in the closure of {x}.

Downward closure. A point x lies in the closure of a subset S of X if and only if there is a point y in Ssuch that x y where is the specialization preorder i.e. x lies in the closure of {y}.

Finite generation and category theoretic characterizations: Finite closure. A point x lies within the closure of a subset S of X if and only if there is a nite subsetF of S such that x lies in the closure of F.

Finite subspace. T is coherent with the nite subspaces of X. Finite inclusion map. The inclusion maps fi : Xi X of the nite subspaces of X form a nal sink. Finite generation. X is nitely generated i.e. it is in the nal hull of the nite spaces. (This means that

there is a nal sink fi : Xi X where each Xi is a nite topological space.)

Topological spaces satisfying the above equivalent characterizations are called nitely generated spaces or Alexan-drov spaces and their topology T is called the Alexandrov topology, named after the Russian mathematician PavelAlexandrov who rst investigated them.

1.2 Duality with preordered sets

1.2.1 The Alexandrov topology on a preordered setGiven a preordered set X = hX;i we can dene an Alexandrov topology on X by choosing the open sets to bethe upper sets:

= fG X : 8x; y 2 X x 2 G ^ x y ! y 2 G; gWe thus obtain a topological space T(X) = hX; i .The corresponding closed sets are the lower sets:

fS X : 8x; y 2 X x 2 S ^ y x ! y 2 S; g

1.2.2 The specialization preorder on a topological spaceGiven a topological space X = the specialization preorder on X is dened by:

xy if and only if x is in the closure of {y}.

We thus obtain a preordered set W(X) = .

1.2.3 Equivalence between preorders and Alexandrov topologiesFor every preordered set X = we always have W(T(X)) = X, i.e. the preorder of X is recovered from thetopological space T(X) as the specialization preorder. Moreover for every Alexandrov space X, we have T(W(X)) =X, i.e. the Alexandrov topology of X is recovered as the topology induced by the specialization preorder.However for a topological space in general we do not have T(W(X)) = X. Rather T(W(X)) will be the set X with aner topology than that of X (i.e. it will have more open sets).

1.2. DUALITY WITH PREORDERED SETS 3

1.2.4 Equivalence between monotony and continuityGiven a monotone function

f : XY

between two preordered sets (i.e. a function

f : XY

between the underlying sets such that xy in X implies f(x)f(y) in Y), let

T(f) : T(X)T(Y)

be the same map as f considered as a map between the corresponding Alexandrov spaces. Then

T(f) : T(X)T(Y)

is a continuous map.Conversely given a continuous map

f : XY

between two topological spaces, let

W(f) : W(X)W(Y)

be the same map as f considered as a map between the corresponding preordered sets. Then

W(f) : W(X)W(Y)

is a monotone function.Thus a map between two preordered sets is monotone if and only if it is a continuous map between the correspondingAlexandrov spaces. Conversely a map between two Alexandrov spaces is continuous if and only if it is a monotonefunction between the corresponding preordered sets.Notice however that in the case of topologies other than the Alexandrov topology, we can have a map between twotopological spaces that is not continuous but which is nevertheless still a monotone function between the correspondingpreordered sets. (To see this consider a non-Alexandrov space X and consider the identity map

i : XT(W(X)).)

1.2.5 Category theoretic description of the dualityLet Set denote the category of sets and maps. Let Top denote the category of topological spaces and continuousmaps; and let Pro denote the category of preordered sets and monotone functions. Then

T : ProTop and

W : TopPro

are concrete functors over Set which are left and right adjoints respectively.Let Alx denote the full subcategory of Top consisting of the Alexandrov spaces. Then the restrictions

4 CHAPTER 1. ALEXANDROV TOPOLOGY

T : ProAlx and

W : AlxPro

are inverse concrete isomorphisms over Set.Alx is in fact a bico-reective subcategory of Top with bico-reector TW : TopAlx. This means that given atopological space X, the identity map

i : T(W(X))X

is continuous and for every continuous map

f : YX

where Y is an Alexandrov space, the composition

i 1f : YT(W(X))

is continuous.

1.2.6 Relationship to the construction of modal algebras from modal framesGiven a preordered set X, the interior operator and closure operator of T(X) are given by:

Int(S) = { x X : for all y X, xy implies y S }, for all S X

Cl(S) = { x X : there exists a y S with xy } for all S X

Considering the interior operator and closure operator to be modal operators on the power set Boolean algebra of X,this construction is a special case of the construction of a modal algebra from a modal frame i.e. a set with a singlebinary relation. (The latter construction is itself a special case of a more general construction of a complex algebrafrom a relational structure i.e. a set with relations dened on it.) The class of modal algebras that we obtain in thecase of a preordered set is the class of interior algebrasthe algebraic abstractions of topological spaces.

1.3 HistoryAlexandrov spaces were rst introduced in 1937 by P. S. Alexandrov under the name discrete spaces, where heprovided the characterizations in terms of sets and neighbourhoods.[1] The name discrete spaces later came to be usedfor topological spaces in which every subset is open and the original concept lay forgotten. With the advancement ofcategorical topology in the 1980s, Alexandrov spaces were rediscovered when the concept of nite generation wasapplied to general topology and the name nitely generated spaces was adopted for them. Alexandrov spaces werealso rediscovered around the same time in the context of topologies resulting from denotational semantics and domaintheory in computer science.In 1966 Michael C. McCord and A. K. Steiner each independently observed a duality between partially ordered setsand spaces which were precisely the T0 versions of the spaces that Alexandrov had introduced.[2][3] P. Johnstonereferred to such topologies as Alexandrov topologies.[4] F. G. Arenas independently proposed this name for thegeneral version of these topologies.[5] McCord also showed that these spaces are weak homotopy equivalent to theorder complex of the corresponding partially ordered set. Steiner demonstrated that the duality is a contravariantlattice isomorphism preserving arbitrary meets and joins as well as complementation.It was also a well known result in the eld of modal logic that a duality exists between nite topological spaces andpreorders on nite sets (the nite modal frames for the modal logic S4). C. Naturman extended these results to aduality between Alexandrov spaces and preorders in general, providing the preorder characterizations as well as theinterior and closure algebraic characterizations.[6]

A systematic investigation of these spaces from the point of view of general topology which had been neglected sincethe original paper by Alexandrov, was taken up by F.G. Arenas.[5]

1.4. SEE ALSO 5

1.4 See also P-space, a space satisfying the weaker condition that countable intersections of open sets are open

1.5 References[1] Alexandro, P. (1937). Diskrete Rume. Mat. Sb. (N.S.) (in German) 2: 501518.

[2] McCord, M. C. (1966). Singular homology and homotopy groups of nite topological spaces. DukeMathematical Journal33 (3): 465474. doi:10.1215/S0012-7094-66-03352-7.

[3] Steiner, A. K. (1966). The Lattice of Topologies: Structure and Complementation. Transactions of the American Math-ematical Society 122 (2): 379398. doi:10.2307/1994555. ISSN 0002-9947.

[4] Johnstone, P. T. (1986). Stone spaces (1st paperback ed.). New York: Cambridge University Press. ISBN 0-521-33779-8.

[5] Arenas, F. G. (1999). Alexandro spaces (PDF). Acta Math. Univ. Comenianae 68 (1): 1725.

[6] Naturman, C. A. (1991). Interior Algebras and Topology. Ph.D. thesis, University of Cape Town Department of Mathe-matics.

Chapter 2

Approach space

In topology, approach spaces are a generalization of metric spaces, based on point-to-set distances, instead of point-to-point distances. They were introduced by Robert Lowen in 1989.

2.1 DenitionGiven a metric space (X,d), or more generally, an extended pseudoquasimetric (which will be abbreviated pq-metrichere), one can dene an induced map d:XP(X)[0,] by d(x,A) = inf { d(x,a ) : a A }. With this example inmind, a distance on X is dened to be a map XP(X)[0,] satisfying for all x in X and A, B X,

1. d(x,{x}) = 0 ;

2. d(x,) = ;

3. d(x,AB) = min d(x,A),d(x,B) ;

4. For all , 0, d(x,A) d(x,A()) + ;

where A() = { x : d(x,A) } by denition.(The empty inmum is positive innity convention is like the nullary intersection is everything convention.)An approach space is dened to be a pair (X,d) where d is a distance function on X. Every approach space has atopology, given by treating A A(0) as a Kuratowski closure operator.The appropriate maps between approach spaces are the contractions. A map f:(X,d)(Y,e) is a contraction ife(f(x),f[A]) d(x,A) for all x X, A X.

2.2 ExamplesEvery pq-metric space (X,d) can be distancized to (X,d), as described at the beginning of the denition.Given a set X, the discrete distance is given by d(x,A) = 0 if x A and = if x A. The induced topology is thediscrete topology.Given a set X, the indiscrete distance is given by d(x,A) = 0 if A is non-empty, and = if A is empty. The inducedtopology is the indiscrete topology.Given a topological space X, a topological distance is given by d(x,A) = 0 if x A, and = if not. The inducedtopology is the original topology. In fact, the only two-valued distances are the topological distances.Let P=[0,], the extended positive reals. Let d+(x,A) = max (xsup A,0) for xP and AP. Given any approachspace (X,d), the maps (for each AX) d(.,A) : (X,d) (P,d+) are contractions.

6

2.3. EQUIVALENT DEFINITIONS 7

On P, let e(x,A) = inf { |xa| : aA } for x0, NA[] .

Given a distance d, the associated AA() is a tower. Conversely, given a tower, the map d(x,A) = inf { : x A[]} is a distance, and these two operations are inverses of each other.A contraction f:(X,d)(Y,e) is, in terms of associated towers, a map such that for all 0, f[A[]] f[A][].

2.4 Categorical propertiesThe main interest in approach spaces and their contractions is that they form a category with good properties, whilestill being quantitative like metric spaces. One can take arbitrary products and coproducts and quotients, and theresults appropriately generalize the corresponding results for topologies. One can even distancize such badly non-metrizable spaces like N, the Stoneech compactication of the integers.Certain hyperspaces, measure spaces, and probabilistic metric spaces turn out to be naturally endowed with a distance.Applications have also been made to approximation theory.

2.5 References Lowen, Robert (1997). Approach spaces: the missing link in the topology-uniformity-metric triad. Oxford

Mathematical Monographs. Oxford: Clarendon Press. ISBN 0-19-850030-0. Zbl 0891.54001. Lowen, Robert (2015). Index Analysis: Approach Theory at Work. Springer.

8 CHAPTER 2. APPROACH SPACE

2.6 External links Robert Lowen

Chapter 3

Baire space

For the concept in set theory, see Baire space (set theory).

In mathematics, a Baire space is a topological space that has enough points that every intersection of a countablecollection of open dense sets in the space is also dense. Complete metric spaces and locally compact Hausdor spacesare examples of Baire spaces according to the Baire category theorem. The spaces are named in honor of Ren-LouisBaire who introduced the concept.

3.1 MotivationIn an arbitrary topological space, the class of closed sets with empty interior consists precisely of the boundaries ofdense open sets. These sets are, in a certain sense, negligible. Some examples are nite sets in , smooth curves inthe plane, and proper ane subspaces in a Euclidean space. If a topological space is a Baire space then it is large,meaning that it is not a countable union of negligible subsets. For example, the three-dimensional Euclidean space isnot a countable union of its ane planes.

3.2 DenitionThe precise denition of a Baire space has undergone slight changes throughout history, mostly due to prevailingneeds and viewpoints. First, we give the usual modern denition, and then we give a historical denition which iscloser to the denition originally given by Baire.

3.2.1 Modern denitionA Baire space is a topological space in which the union of every countable collection of closed sets with emptyinterior has empty interior.This denition is equivalent to each of the following conditions:

Every intersection of countably many dense open sets is dense. The interior of every union of countably many closed nowhere dense sets is empty. Whenever the union of countably many closed subsets of X has an interior point, then one of the closed subsets

must have an interior point.

3.2.2 Historical denitionMain article: Meagre set

9

10 CHAPTER 3. BAIRE SPACE

In his original denition, Baire dened a notion of category (unrelated to category theory) as follows.A subset of a topological space X is called

nowhere dense in X if the interior of its closure is empty of rst category or meagre in X if it is a union of countably many nowhere dense subsets of second category or nonmeagre in X if it is not of rst category in X

The denition for a Baire space can then be stated as follows: a topological spaceX is a Baire space if every non-emptyopen set is of second category in X. This denition is equivalent to the modern denition.A subset A of X is comeagre if its complement X nA is meagre. A topological space X is a Baire space if and onlyif every comeager subset of X is dense.

3.3 Examples The space R of real numbers with the usual topology, is a Baire space, and so is of second category in itself.

The rational numbers are of rst category and the irrational numbers are of second category in R . The Cantor set is a Baire space, and so is of second category in itself, but it is of rst category in the interval[0; 1] with the usual topology.

Here is an example of a set of second category in R with Lebesgue measure 0.

1\m=1

1[n=1

rn 1

2n+m; rn +

1

2n+m

where frng1n=1 is a sequence that enumerates the rational numbers.

Note that the space of rational numbers with the usual topology inherited from the reals is not a Baire space,since it is the union of countably many closed sets without interior, the singletons.

3.4 Baire category theoremMain article: Baire category theorem

The Baire category theorem gives sucient conditions for a topological space to be a Baire space. It is an importanttool in topology and functional analysis.

(BCT1) Every complete metric space is a Baire space. More generally, every topological space which ishomeomorphic to an open subset of a complete pseudometric space is a Baire space. In particular, everycompletely metrizable space is a Baire space.

(BCT2) Every locally compact Hausdor space (or more generally every locally compact sober space) is aBaire space.

BCT1 shows that each of the following is a Baire space:

The space R of real numbers The space of irrational numbers, which is homeomorphic to the Baire space of set theory The Cantor set Indeed, every Polish space

BCT2 shows that every manifold is a Baire space, even if it is not paracompact, and hence not metrizable. Forexample, the long line is of second category.

3.5. PROPERTIES 11

3.5 Properties Every non-empty Baire space is of second category in itself, and every intersection of countably many dense

open subsets of X is non-empty, but the converse of neither of these is true, as is shown by the topologicaldisjoint sum of the rationals and the unit interval [0, 1].

Every open subspace of a Baire space is a Baire space.

Given a family of continuous functions fn:XY with pointwise limit f:XY. If X is a Baire space then thepoints where f is not continuous is a meagre set in X and the set of points where f is continuous is dense in X.A special case of this is the uniform boundedness principle.

A closed subset of a Baire space is not necessarily Baire.

The product of two Baire spaces is not necessarily Baire. However, there exist sucient conditions that willguarantee that a product of arbitrarily many Baire spaces is again Baire.

3.6 See also BanachMazur game Descriptive set theory Baire space (set theory)

3.7 References

3.8 Sources Munkres, James, Topology, 2nd edition, Prentice Hall, 2000. Baire, Ren-Louis (1899), Sur les fonctions de variables relles, Annali di Mat. Ser. 3 3, 1123.

3.9 External links Encyclopaedia of Mathematics article on Baire space Encyclopaedia of Mathematics article on Baire theorem

Chapter 4

Baire space (set theory)

For the concept in topology, see Baire space.

In set theory, the Baire space is the set of all innite sequences of natural numbers with a certain topology. Thisspace is commonly used in descriptive set theory, to the extent that its elements are often called reals. It is oftendenoted B, NN, , or . Moschovakis denotes it N .The Baire space is dened to be the Cartesian product of countably innitely many copies of the set of naturalnumbers, and is given the product topology (where each copy of the set of natural numbers is given the discretetopology). The Baire space is often represented using the tree of nite sequences of natural numbers.The Baire space can be contrasted with Cantor space, the set of innite sequences of binary digits.

4.1 Topology and treesThe product topology used to dene the Baire space can be described more concretely in terms of trees. The denitionof the product topology leads to this characterization of basic open sets:

If any nite set of natural number coordinates {ci : i < n } is selected, and for each ci a particular naturalnumber value vi is selected, then the set of all innite sequences of natural numbers that have value viat position ci for all i < n is a basic open set. Every open set is a union of a collection of these.

By moving to a dierent basis for the same topology, an alternate characterization of open sets can be obtained:

If a sequence of natural numbers {wi : i < n} is selected, then the set of all innite sequences of naturalnumbers that have value wi at position i for all i < n is a basic open set. Every open set is a union of acollection of these.

Thus a basic open set in the Baire space species a nite initial segment of an innite sequence of natural numbers,and all the innite sequences extending form a basic open set. This leads to a representation of the Baire space asthe set of all paths through the full tree

4.3. RELATION TO THE REAL LINE 13

1. It is a perfect Polish space, which means it is a completely metrizable second countable space with no isolatedpoints. As such, it has the same cardinality as the real line and is a Baire space in the topological sense of theterm.

2. It is zero-dimensional and totally disconnected.3. It is not locally compact.

4. It is universal for Polish spaces in the sense that it can be mapped continuously onto any non-empty Polishspace. Moreover, any Polish space has a dense G subspace homeomorphic to a G subspace of the Bairespace.

5. The Baire space is homeomorphic to the product of any nite or countable number of copies of itself.

4.3 Relation to the real lineThe Baire space is homeomorphic to the set of irrational numbers when they are given the subspace topology inheritedfrom the real line. A homeomorphism between Baire space and the irrationals can be constructed using continuedfractions.From the point of view of descriptive set theory, the fact that the real line is connected causes technical diculties.For this reason, it is more common to study Baire space. Because every Polish space is the continuous image of Bairespace, it is often possible to prove results about arbitrary Polish spaces by showing that these properties hold for Bairespace and by showing that they are preserved by continuous functions.B is also of independent, but minor, interest in real analysis, where it is considered as a uniform space. The uniformstructures ofB and Ir (the irrationals) are dierent, however: B is complete in its usual metric while Ir is not (althoughthese spaces are homeomorphic).

4.4 References Kechris, Alexander S. (1994). Classical Descriptive Set Theory. Springer-Verlag. ISBN 0-387-94374-9. Moschovakis, Yian

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