Foundations of General Topology (eBook)

Front Cover 1
Foundations of General Topology 4
Copyright Page 5
Table of Contents 10
Dedication 6
Preface 8
Chapter 1. Algebra of Sets 14
1.1 Sets and Subsets 14
1.2 Operations on Sets 17
1.3 Relations 19
1.4 Mappings 23
1.5 Partial Orders 27
Chapter 2. Cardinal and Ordinal Numbers 30
2.1 Equipotent Sets 30
2.2 Cardinal Numbers 34
2.3 Order Types 39
2.4 Ordinal Numbers 42
2.5 Axiom of Choice 46
Chapter 3. Topological Spaces 49
Introduction 49
3.1 Open Sets and Limit Points 49
3.2 Closed Sets and Closure 52
3.3 Operators and Neighborhoods 56
3.4 Bases and Relative Topologies 59
Chapter 4. Connectedness, Compactness, and Continuity 64
4.1 Connected Sets and Components 64
4.2 Compact and Countably Compact Spaces 69
4.3 Continuous Functions 73
4.4 Homeomorphisms 77
4.5 Arcwise Connectivity 80
Chapter 5. Separation and Countability Axioms 82
5.1 T0- and T1-Spaces 82
5.2 T2-Spaces and Sequences 86
5.3 Axioms of Countability 92
5.4 Separability and Summary 97
5.5 Regular and Normal Spaces 100
5.6 Completely Regular Spaces 108
Chapter 6. Metric Spaces 112
6.1 Metric Spaces as Topological Spaces 112
6.2 Topological Properties 117
6.3 Hilbert (I2) Space 121
6.4 Fréchet Space 125
6.5 Space of Continuous Functions 128
Chapter 7. Complete Metric Spaces 131
7.1 Cauchy Sequences 131
7.2 Completions 135
7.3 Equivalent Conditions 138
7.4 Baire Theorem 140
Chapter 8. Product Spaces 142
8.1 Finite Products 142
8.2 Product Invariant Properties 145
8.3 Metric Products 148
8.4 Tichonov Topology 151
8.5 Tichonov Theorem 155
Chapter 9. Function and Quotient Spaces 160
9.1 Topology of Pointwise Convergence 160
9.2 Topology of Compact Convergence 162
9.3 Quotient Topology 166
Chapter 10. Metrization and Paracompactness 171
10.1 Urysohn's Metrization Theorem 171
10.2 Paracompact Spaces 173
10.3 Nagata-Smirnov Metrization Theorem 182
Chapter 11. Uniform Spaces 187
11.1 Quasi Uniformization 187
11.2 Uniformization 191
11.3 Uniform Continuity 197
11.4 Completeness and Compactness 201
11.5 Proximity Spaces 206
Bibliography 210
Index 214