목차 일부
CONTENTS
Preface ... ⅲ
Chapter 0 Set Theory ... 1
0.1 Sets ... 1
0.2 Relations and Functions ... 6
0.3 Equivalence Relations ... 10
0.4 Order Relations ... 12
0.5 The Integers and the ...
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목차 전체
CONTENTS
Preface ... ⅲ
Chapter 0 Set Theory ... 1
0.1 Sets ... 1
0.2 Relations and Functions ... 6
0.3 Equivalence Relations ... 10
0.4 Order Relations ... 12
0.5 The Integers and the Real Numbers ... 14
0.6 Maximality and Zorns Lemma ... 16
0.7 Cardinality of Sets ... 18
Chapter 1 Pseudometric Spaces ... 23
1.1 Pseudometrics ... 23
1.2 Closed and Open Sets ... 27
Chapter 2 Topological Spaces ... 31
2.1 Topological Spaces ... 31
2.2 Base for a Topology ... 39
2.3 Subspaces ... 43
Chapter 3 Continuous Functions ... 47
3.1 Continuity of a Function ... 47
3.2 Homeomorphisms ... 53
3.3 The Weak Topology by a Collection of Functions ... 57
Chapter 4 Connected Spaces ... 63
4.1 Connected Spaces ... 63
4.2 Components of a Space ... 68
4.3 Path-Connected Spaces ... 70
4.4 Locally Connected and Locally Path-Connected Spaces ... 72
Chapter 5 Compact Spaces ... 77
5.1 Compact Spaces ... 77
5.2 The One-Point Compactification ... 85
Chapter 6 Product Spaces ... 89
6.1 Products of Spaces ... 89
6.2 Continuous Functions and Slices in Product Spaces ... 93
6.3 Products of Hausdorff Spaces; Products of Connected Spaces ... 95
6.4 Products of Compact Spaces ... 97
6.5 Products of Pseudometric Spaces ... 99
6.6 The Hilbert Cube ... 101
6.7 The Cantor Space ... 104
Chapter 7 Sequences ... 111
7.1 Sequences ... 111
7.2 Sequences and Compact Spaces ... 117
7.3 Nets ... 122
Chapter 8 Complete Pseudometric Spaces ... 127
8.1 Cauchy Sequences and Complete Spaces ... 127
8.2 Baire Category Theorem ... 132
8.3 Uniform Continuity ... 134
8.4 Completion of a Pseudometric Space ... 136
8.5 Banach Fixed-Point Theorem ... 139
Chapter 9 Euclidean Spaces ... 143
9.1 Euclidean n-Space ... 143
9.2 Space-Filling Curves ... 149
9.3 Pseudonorms ... 152
9.4 Spheres ... 155
Chapter 10 Quotient Spaces ... 159
10.1 The Strong Topology and the Quotient Topology ... 159
10.2 Quotient Maps ... 162
10.3 Quotient Spaces ... 165
10.4 The Metric Identification ... 168
Chapter 11 Hyperspaces and Multifunctions ... 171
11.1 Hyperspaces ... 171
11.2 Quotient Spaces and Hyperspaces ... 175
11.3 The Hausdorff Metric ... 179
11.4 Multifunctions ... 186
11.5 Functions Induced by Multifunctions ... 190
Chapter 12 Dimension ... 195
12.1 Topological Dimension ... 195
12.2 Dimension of Subspaces ... 198
12.3 Dimension in Rn ... 201
12.4 Hausdorff Dimension ... 202
References ... 209
Index ... 211
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