Contents - unito.itwebmath2.unito.it/paginepersonali/sergio.console/Dispense/topology... · *36...
Transcript of Contents - unito.itwebmath2.unito.it/paginepersonali/sergio.console/Dispense/topology... · *36...
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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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Contents v
. . . . . . . . . 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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