Introduction to Topology: Pure and Applied

508 Pages·2007·16.57 MB·English

by  Colin Adams

INTRODUCTION TO
TOPOLOGY
PURE AND APPLIED
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Cohn's Dedication:
To Ahexa and Colton
Bob's Dedication:
Ahla Famighia

Contents
Preface 11
o Introduction 19
0.1 What is Topology and How is it Applied? 19
0.2 A Glimpse at the History 25
0.3 Sets and Operations on Them 27
0.4 Euclidean Space 32
0.5 Relations 35
0.6 Functions 37
1 Topological Spaces 42
1.1 Open Sets and the Definition of a Topology 42
1.2 Basis for a Topology 46
1.3 Closed Sets 56
1.4 Examples of Topologies in Applications 61
2 Interior, Closure, and Boundary 73
2.1 Interior and Closure of Sets 73
2.2 Limit Points 78
2.3 The Boundary of a Set 83
2.4 An Application to Geographic Information Systems 86
3 Creating New Topological Spaces 94
3.1 The Subspace Topology 94
3.2 The Product Topology 100
3.3 The Quotient Topology 106
3.4 More Examples of Quotient Spaces 115
3.5 Configuration Spaces and Phase Spaces 123
4 Continuous Functions and Homeomorphisms 130
4.1 Continuity 130
4.2 Homeomorphisms 140
4.3 The Forward Kinematics Map in Robotics 153
S Metric Spaces 161
5.1 Metrics 161
5.2 Metrics and Information 168
5.3 Properties of Metric Spaces 173
5.4 Metrizability 180
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Contents
6 Connectedness 186
6.1 A First Approach to Connectedness 186
• . . .
6.2 Distinguishing Topological Spaces via Connectedness. 196
• . . .
6.3 The Intermediate Value Theorem 203
• . . .
6.4 Path Connectedness 209
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6.5 Automated Guided Vehicles 215
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7 Compactness 223
7.1 Open Coverings and Compact Spaces 223
7.2 Compactness in Metric Spaces 231
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7.3 The Extreme Value Theorem 238
7.4 Limit Point Compactness 245
7.5 One-Point Compactifications 250
8 Dynamical Systems and Chaos 256
8.1 Iterating Functions 256
8.2 Stability 264
8.3 Chaos 272
8.4 A Simple Population Model with Complicated Dynamics . 281
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8.5 Chaos Implies Sensitive Dependence on Initial Conditions 287
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9 Homotopy and Degree Theory 291
9.1 Homotopy 291
9.2 Circle Functions, Degree, and Retractions 295
9.3 An Application to a Heartbeat Model. 302
9.4 The Fundamental Theorem of Algebra 306
9.5 More on Distinguishing Topological Spaces 308
9.6 More on Degree 313
10 Fixed Point Theorems and Applications 323
10.1 The Brouwer Fixed Point Theorem 323
10.2 An Application to Economics 328
10.3 Kakutani's Fixed Point Theorem 336
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10.4 Game Theory and the Nash Equilibrium 343
11 Embeddings 351
11.1 Some Embedding Results 351
11.2 The Jordan Curve Theorem 359
11.3 Digital Topology and Digital Image Processing 366
12 Knots 376
12.1 Isotopy and Knots 377
12.2 Reidemeister Moves and Linking Number 386
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12.3 Polynomialsof Knots 393
12.4 Applications to Biochemistry and Chemistry. 399
Contents 9
13 Graphs and Topology 408
13.1 Graphs 408
13.2 Chemical Graph Theory 416
13.3 Graph Embeddings 422
13.4 Crossing Number and Thickness 431
14 Manifolds and Cosmology 441
14.1 Manifolds 442
14.2 Euler Characteristic and the Classification of Compact Surfaces 454
14.3 Three-Manifolds 464
14.4 The Geometry of the Universe 477
14.5 Determining which Manifold is the Universe 481
Additional Readings 485
References 489
Index 497