Topology

James Munkres9780131816299, 0131816292

This introduction to topology provides separate, in-depth coverage of both general topology and algebraic topology. Includes many examples and figures. GENERAL TOPOLOGY. Set Theory and Logic. Topological Spaces and Continuous Functions. Connectedness and Compactness. Countability and Separation Axioms. The Tychonoff Theorem. Metrization Theorems and paracompactness. Complete Metric Spaces and Function Spaces. Baire Spaces and Dimension Theory. ALGEBRAIC TOPOLOGY. The Fundamental Group. Separation Theorems. The Seifert-van Kampen Theorem. Classification of Surfaces. Classification of Covering Spaces. Applications to Group Theory. For anyone needing a basic, thorough, introduction to general and algebraic topology and its applications.

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Table of contents :
Cover ……Page 1
Title page ……Page 2
Date-line ……Page 3
Dedication ……Page 4
Contents ……Page 6
Preface ……Page 10
A Note to the Reader ……Page 14
Part I. GENERAL TOPOLOGY ……Page 16
Chapter 1. Set Theory and Logic ……Page 18
1 Fundamental Concepts ……Page 19
2 Functions ……Page 30
3 Relations ……Page 36
4 The Integers and the Real Numbers ……Page 45
5 Cartesian Products ……Page 51
6 Finite Sets ……Page 54
7 Countable and Uncountable Sets ……Page 59
*8 The Principle of Recursive Definition ……Page 67
9 Infinite Sets and the Axiom of Choice ……Page 72
10 Well-Ordered Sets ……Page 77
*11 The Maximum Principle ……Page 83
*Supplementary Exercises: Well-Ordering ……Page 87
12 Topological Spaces ……Page 90
13 Basis for a Topology ……Page 93
14 The Order Topology ……Page 99
15 The Product Topology on $Xtimes Y$ ……Page 101
16 The Subspace Topology ……Page 103
17 Closed Sets and Limit Points ……Page 107
18 Continuous Functions ……Page 117
19 The Product Topology ……Page 127
20 The Metric Topology ……Page 134
21 The Metric Topology (continued) ……Page 144
*22 The Quotient Topology ……Page 151
*Supplementary Exercises: Topological Groups ……Page 160
Chapter 3. Connectedness and Compactness ……Page 162
23 Connected Spaces ……Page 163
24 Connected Subspaces of the Real Line ……Page 168
*25 Components and Local Connectedness ……Page 174
26 Compact Spaces ……Page 178
27 Compact Subspaces of the Real Line ……Page 187
28 Limit Point Compactness ……Page 193
29 Local Compactness ……Page 197
*Supplementary Exercises: Nets ……Page 202
Chapter 4. Countability and Separation Axioms ……Page 204
30 The Countability Axioms ……Page 205
31 The Separation Axioms ……Page 210
32 Normal Spaces ……Page 215
33 The Urysohn Lemma ……Page 222
34 The Urysohn Metrization Theorem ……Page 229
*35 The Tietze Extension Theorem ……Page 234
*36 Imbeddings of Manifolds ……Page 239
*Supplementary Exercises: Review of the Basics ……Page 243
37 The Tychonoff Theorem ……Page 245
38 The Stone-Cech Compactification ……Page 252
Chapter 6. Metrization Theorems and Paracompactness ……Page 258
39 Local Finiteness ……Page 259
40 The Nagata-Smirnov Metrization Theorem ……Page 263
41 Paracompactness ……Page 267
42 The Smirnov Metrization Theorem ……Page 276
Chapter 7. Complete Metric Spaces and Function Spaces ……Page 278
43 Complete Metric Spaces ……Page 279
*44 A Space-Filling Curve ……Page 286
45 Compactness in Metric Spaces ……Page 290
46 Pointwise and Compact Convergence ……Page 296
47 Ascoli’s Theorem ……Page 305
Chapter 8. Baire Spaces and Dimension Theory ……Page 309
48 Baire Spaces ……Page 310
*49 A Nowhere-Differentiable Function ……Page 315
50 Introduction to Dimension Theory ……Page 319
*Supplementary Exercises: Locally Euclidean Spaces ……Page 331
Part II. ALGEBRAIC TOPOLOGY ……Page 334
Chapter 9. The Fundamental Group ……Page 336
51 Homotopy of Paths ……Page 337
52 The Fundamental Group ……Page 345
53 Covering Spaces ……Page 350
54 The Fundamental Group of the Circle ……Page 356
55 Retractions and Fixed Points ……Page 363
*56 The Fundamental Theorem of Algebra ……Page 368
*57 The Borsuk-Ulam Theorem ……Page 371
58 Deformation Retracts and Homotopy Type ……Page 374
59 The Fundamental Group of $S^n$ ……Page 383
60 Fundamental Groups of Some Surfaces ……Page 385
61 The Jordan Separation Theorem ……Page 391
*62 Invariance of Domain ……Page 396
63 The Jordan Curve Theorem ……Page 400
64 Imbedding Graphs in the Plane ……Page 409
65 The Winding Number of a Simple Closed Curve ……Page 413
66 The Cauchy Integral Formula ……Page 418
67 Direct Sums of Abelian Groups ……Page 422
68 Free Products of Groups ……Page 427
69 Free Groups ……Page 436
70 The Seifert-van Kampen Theorem ……Page 441
71 The Fundamental Group of a Wedge of Circles ……Page 449
72 Adjoining a Two-cell ……Page 453
73 The Fundamental Groups of the Torus and the Dunce Cap ……Page 457
74 Fundamental Groups of Surfaces ……Page 461
75 Homology of Surfaces ……Page 469
76 Cutting and Pasting ……Page 472
77 The Classification Theorem ……Page 477
78 Constructing Compact Surfaces ……Page 486
Chapter 13. Classification of Covering Spaces ……Page 492
79 Equivalence of Covering Spaces ……Page 493
80 The Universal Covering Space ……Page 499
*81 Covering Transformations ……Page 502
82 Existence of Covering Spaces ……Page 509
*Supplementary Exercises: Topological Properties and $pi_1$ ……Page 514
83 Covering Spaces of a Graph ……Page 516
84 The Fundamental Group of a Graph ……Page 521
85 Subgroups of Free Groups ……Page 528
Bibliography ……Page 532
Index ……Page 534