Introduction 1
1．What is point-set topology about? 1
2．Origin and beginnings 2
CHAPTER Ⅰ Fundamental Concepts 5
1．Theconcept of a topological space 5
2．Metric spaces 7
3．Subspaces,disjoint unions and products 9
4．Bases and subbases 12
5．Continuous maps 12
6．Connectedness 14
7．The Hausdorff separation axiom 17
8．Compactness 18
CHAPTER Ⅱ Topological Vector Spaces 24
1．The notion of a topological vector space 24
2．Finite-dimensional vector spaces 25
3．Hilbert spaces 26
4．Banach spaces 26
5．Fréchet spaces 27
6．Locally convex topological vector spaces 28
7．A couple of examples 29
CHAPTER Ⅲ The Quotient Topology 31
1．The notion of a quotient space 31
2．Quotients and maps 32
3．Properties of quotient spaces 33
4．Examples:Homogeneous spaces 34
5．Examples:Orbit spaces 37
6．Examples:Collapsing a subspace to a point 39
7．Examples:Gluing topological spaces together 43
CHAPTER Ⅳ Completion of Metric Spaces 50
1．The completion of a metric space 50
2．Completion of a map 54
3．Completion of normed spaces 55
CHAPTER Ⅴ Homotopy 59
1．Homotopic maps 59
2．Homotopy equivalence 61
3．Examples 63
4．Categories 66
5．Functors 69
6．What is algebraic topology? 70
7．Homotopy-what for? 74
CHAPTER Ⅵ The Two Countability Axioms 78
1．First and second countability axioms 78
2．Infinite products 80
3．The role of the countability axioms 81
CHAPTER Ⅶ CW-Complexes 87
1．Simplicial complexes 87
2．Cell decompositions 93
3．The notion of a CW-complex 95
4．Subcomplexes 98
5．Cell attaching 99
6．Why CW-complexes are more flexible 100
7．Yes,but...? 102
CHAPTER Ⅷ Construction of Continuous Functions on Topological Spaces 106
1．The Urysohn lemma 106
2．The proof of the Urysohn lemma 111
3．The Tietze extension lemma 114
4．Partitions of unity and vector bundle sections 116
5．Paracompactness 123
CHAPTER Ⅸ Covering Spaces 127
1．Topological spaces over Ⅹ 127
2．The concept of a covering space 130
3．Path lifting 133
4．Introduction to the classification of covering spaces 137
5．Fundamental group and lifting behavior 141
6．The classification of covering spaces 144
7．Covering transformations and universal cover 149
8．The role of covering spaces in mathematics 156
CHAPTER Ⅹ The Theorem of Tychonoff 160
1．An unlikely theorem? 160
2．What is it good for? 162
3．The proof 167
LAST CHAPTER Set Theory(by Theodor Br？cker） 171
References 177
Table of Symbols 179
Index 183