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Contents Preface ...................................... vii ..... A Note to the Reader. ............................... xi Part I GENERAL TOPOLOGY Chapter 1 Set Theory and Logic ....................... 3 1 Fundamental Concepts .......................... 4 2 Functions.. ............................... 15 L , . + 3 Relations ................................. 21 4 The Integers and the Real Numbers ................... 30 5 Cartesian Products ............................ 36 6 Finitesets ................................ 39 7 Countable and Uncountable Sets .................... 44 *8 The Principle of Recursive Definition .................. 52 h F 9 Infinite Sets and the Axiom of Choice .................. 57 10 Well-ordered Sets ............................ 62 * 1 1 The Maximum Principle ......................... 68 *Supplementary Exercises: Well-Ordering ................... 72

Transcript of Contents€¦ · iv Con tents Chapter 2 Topological Spaces and Continuous Functions .....75 12...

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Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii . . . . .

A Note to the Reader. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

Part I GENERAL TOPOLOGY

Chapter 1 Set Theory and Logic . . . . . . . . . . . . . . . . . . . . . . . 3 1 Fundamental Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Functions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

L ,.+ 3 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4 The Integers and the Real Numbers . . . . . . . . . . . . . . . . . . . 30 5 Cartesian Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 6 Finitesets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 7 Countable and Uncountable Sets . . . . . . . . . . . . . . . . . . . . 44

*8 The Principle of Recursive Definition . . . . . . . . . . . . . . . . . . 52 h F 9 Infinite Sets and the Axiom of Choice . . . . . . . . . . . . . . . . . . 57

10 Well-ordered Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 * 1 1 The Maximum Principle . . . . . . . . . . . . . . . . . . . . . . . . . 68 *Supplementary Exercises: Well-Ordering . . . . . . . . . . . . . . . . . . . 72

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iv Con tents

. . . . . . . . . Chapter 2 Topological Spaces and Continuous Functions 75 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Topological Spaces 75

. . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Basis for a Topology 78

. . . . . . . . . . . . . . . . . . . . . . . . . . . 14 The Order Topology 84 . . . . . . . . . . . . . . . . . . . . 15 The Product Topology on X x Y 86

. . . . . . . . . . . . . . . . . . . . . . . . . 16 The Subspace Topology 88 . . . . . . . . . . . . . . . . . . . . . . 17 Closed Sets and Limit Points 92

. . . . . . . . . . . . . . . . . . . . . . . . . . 18 Continuous Functions 102

. . . . . . . . . . . . . . . . . . . . . . . . . . 19 The Product Topology 112 . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 The Metric Topology 119

. . . . . . . . . . . . . . . . . . . . 2 1 The Metric Topology (continued) 129 . . . . . . . . . . . . . . . . . . . . . . . . . *22 The Quotient Topology 136

. . . . . . . . . . . . . . . . *Supplementary Exercises: Topological Groups 145

. . . . . . . . . . . . . . . . . Chapter 3 Connectedness and Compactness 147 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Connected Spaces 148

. . . . . . . . . . . . . . . . . 24 Connected Subspaces of the Real Line 153

. . . . . . . . . . . . . . . . . "25 Components and Local Connectedness 159 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Compact Spaces 163

. . . . . . . . . . . . . . . . . . 27 Compact Subspaces of the Real Line 172 . . . . . . . . . . . . . . . . . . . . . . . . 28 Limit point compactnes$ 178

. . . . . . . . . . . . . . . . . . . . . . . . . . . 29 LocalCompactness 182 . . . . . . . . . . . . . . . . . . . . . . . . *Supplementary Exercises: Nets 187

. . . . . Chapter 4 Countability and Separation Axioms ..I . . . . . . . . . . . . . . . . . . . . . . . . 30 The Countability Axioms 190

. . . . . . . . . . . . . . . . . . . . . . . . . 3 1 The Separation Axioms 195 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Normal Spaces 200

. . . . . . . . . . . . . . . . . . . . . . . . . . . 33 The Urysohn Lemma 207 . . . . . . . . . . . . . . . . . . . 34 The Urysohn Metrization Theorem 214

. . . . . . . . . . . . . . . . . . . . . *35 The Tietze Extension Theorem 219 . . . . . . . . . . . . . . . . . . . . . . . . *36 Imbeddings of Manifolds 224

. . . . . . . . . . . . . . . *Supplementary Exercises: Review of the Basics 228

. . . . . . . . . . . . . . . . . . . . . . Chapter 5 The Tychonoff Theorem 230 . . . . . . . . . . . . . . . . . . . . . . . . . 37 The Tychonoff Theorem 230

. . . . . . . . . . . . . . . . . . . . 38 The stone-cech Compactification 237

. . . . . . . . . . Chapter 6 Metrization Theorems and Paracompactness 243 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Local Finiteness 244

. . . . . . . . . . . . . . . 40 The Nagata-Smirnov Metrization Theorem 248 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Paracompactness 252

. . . . . . . . . . . . . . . . . . . 42 The Smirnov Metrization Theorem 261

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. . . . . . . . . Chapter 7 Complete Metric Spaces and Function Spaces 263 . . . . . . . . . . . . . . . . . . . . . . . . . 43 Complete Metric Spaces 264

. . . . . . . . . . . . . . . . . . . . . . . . . . *44 A Space-Filling Curve 271 . . . . . . . . . . . . . . . . . . . . . . 45 Compactness in Metric Spaces 275

. . . . . . . . . . . . . . . . . . 46 Pointwise and Compact Convergence 281 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Ascoli's Theorem 290

. . . . . . . . . . . . . . . Chapter 8 Baire Spaces and Dimension Theory 294 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Baire Spaces 295

. . . . . . . . . . . . . . . . . . . *49 A Nowhere-Differentiable Function 300 . . . . . . . . . . . . . . . . . . . . 50 Introduction to Dimension Theory 304

. . . . . . . . . . . . . *Supplementary Exercises: Locally Euclidean Spaces 316

Part I1 ALGEBRAIC TOPOLOGY. ii4 E.V . 1

. . . . . . . . . . . . . . . . . . . . . Chapter 9 The Fundamental Group 321 . . . . . . . . . . . . . . . . . . . . . . . . . . . . I 5 1 Homotopy of Paths 322

. . . . . . . . . . . . . . . . . . . . . . . . . 52 The Fundamental Group 330 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Covering Spaces 335

. . . . . . . . . . . . . . . . . . 54 The Fundamental Group of the Circle 341 . . . . . . . . . . . . . . . . . . . . . . . 55 Retractions and Fixed Points 348

. . . . . . . . . . . . . . . . . *56 The Fundamental Theorem of Algebra 353 . . . . . . . . . . . . . . . . . . . . . . . *57 The Borsuk-Ulam Theorem 356

. . . . . . . . . . . . . . . 58 Deformation Retracts and Homotopy Type 359 . . . . . . . . . . . . . . . . . . . . . . 59 The Fundamental Group of Sn 368

. . . . . . . . . . . . . . . . . 60 Fundamental Groups of Some Surfaces 370

. . . . . . . . . . . . . . . Chapter 10 Separation Theorems in the Plane 376 . . . . . . . . . . . . . . . . . . . . . 61 The Jordan Separation Theorem 376

. . . . . . . . . . . . . . . . . . . . . . . . . . *62 Invariance of Domain 381 . . . . . . . . . . . . . . . . . . . . . . . 63 The Jordan Curve Theorem 385

. . . . . . . . . . . . . . . . . . . . . 64 Imbedding Graphs in the Plane 394 . . . . . . . . . . . . 65 The Winding Number of a Simple Closed Curve 398

. . . . . . . . . . . . . . . . . . . . . . 66 The Cauchy Integral Formula 403

. . . . . . . . . . . . . . . Chapter 11 The Seifert-van Kampen Theorem . . . . . . . . . . . . . . . . . . . . . 67 Direct Sums of Abelian Groups

. . . . . . . . . . . . . . . . . . . . . . . . . 68 Free Products of Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Free Groups

. . . . . . . . . . . . . . . . . . . . 70 The Seifert-van Kampen Theorem . . . . . . . . . . . . . 7 1 The Fundamental Group of a Wedge of Circles

. . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Adjoining a Two-cell 73 The Fundamental Groups of the Torus and the Dunce Cap . . . . . . .

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vi Contents

* " s" . . . . . . . . . . . . . . . . . . . Chapter 12 Classification of surfaces" 4%5 . . . . . . . . . . . . . . . . . . . . 74 Fundamental Groups of Surfaces 446

75 Homology of Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 454 . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Cutting and Pasting 457

. . . . . . . . . . . . . . . . . . . . . . . 77 The Classification Theorem 462 . . . . . . . . . . . . . . . . . . . . . 78 Constructing Compact Surfaces 47 1

fl .> . . . . . . . . . . . . . . . . . Chapter 13 Classification of Covering Spa- 477 . . . . . . . . . . . . . . . . . . . . . 79 Equivalence of Covering Spaces 478

. . . . . . . . . . . . . . . . . . . . . . 80 The Universal Covering Space 484 . . . . . . . . . . . . . . . . . . . . . . . . i *8 1 Covering Transformations 487

I 82 Existence of Covering Spaces . . . . . . . . . . . . . . . . . . . . . . 494 *Supplementary Exercises: Topological Properties and rcl . . . . . . . . . . 499

3T.nj!'P. ."s?k!A . . 501 Chapter 14 Applications to Group Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Covering Spaces of a Graph 501

. . . . . . . . . . . . . . . . . . . 84 The Fundamental Group of a Graph 506 . . . . . . . . . . . . . . . . . . . . . . . . 85 Subgroups of Free Groups 5 13

* 2 i J i k , I

> Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519

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Luca Motto Ros - uniroma1.it · (Classical) Descriptive Set Theory Polish spaces= separable completely metrizable topological spaces. Example: 2! with the product topology (which

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Metrizability of Topological Spaces

Metrizability of Topological Spaces

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ACLT: Algebra, Categories, Logic in Topology ...ACLT: Algebra, Categories, Logic in Topology - Grothendieck's generalized topological spaces (toposes) Steve Vickers CS Theory Group