1 Set and Map 1
1.1 Set 2
1.2 Map 7
1.3 Counting 14
1.4 Equivalence Relation and Quotient 19
2 Metric Space 25
2.1 Metric 26
2.2 Ball 30
2.3 Open Subset 32
2.4 Continuity 38
2.5 Limit Point 43
2.6 Closed Subset 45
3 Graph and Network 49
3.1 Seven Bridges in K？nigsberg 50
3.2 Proof of One-Trip Criterion 53
3.3 Euler Formula 56
3.4 Application of Euler Formula 58
4 Topology 63
4.1 Topological Basis and Subbasis 64
4.2 Open Subset 67
4.3 Topological Space 71
4.4 Comparing Topologies 76
4.5 Limit Point and Closed Subset 79
4.6 Closure 84
5 Basic Topological Concepts 89
5.1 Continuity 90
5.2 Homeomorphism 95
5.3 Subspace 99
5.4 Product 103
5.5 Quotient 107
6 Complex 115
6.1 Simplicial Complex 116
6.2 CW-Complex 120
6.3 Projective Space 122
6.4 Euler Number 125
7 Topological Properties 129
7.1 Hansdorff Space 130
7.2 Connected Space 133
7.3 Path Connected Space 137
7.4 Connected Component 143
7.5 Compact Space 145
7.6 Limit Point Compact Space 156
8 Surface 161
8.1 Manifold 162
8.2 Surface 164
8.3 Simplicial Surface 168
8.4 Planar Diagram 170
8.5 Cut and Paste 173
8.6 Classification of Surface 176
8.7 Recognition of Surface 179
9 Topics in Point Set Topology 187
9.1 Normal Space 188
9.2 Paracompact Space 192
9.3 Complete Metric Space 196
9.4 Baire Category Theorem 200
9.5 Infinite Product 207
9.6 Space-Filling Curve 214
9.7 Space of Maps 218
Index 229