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Introduction,a Survey of Some Topological Concepts 3

1．Theory of Sets．Topological Spaces 3

2．Questions Related to Curves 5

3．Polyhedra 8

4．Coincidences and Fixed Points 14

5．Vector Fields 17

6．Integration and Topology 19

Chapter Ⅰ．Basic Information about Sets,Spaces,Vectors,Groups 26

1．Questions of Notation and Terminology 26

2．Euclidean Spaces,Metric Spaces,Topological Spaces 28

3．Compact Spaces 34

4．Vector Spaces 38

5．Products of Sets,Spaces and Groups．Homotopy 40

Problems 43

Chapter Ⅱ．Two-dimensional Polyhedral Topology 45

1．Elements of the Theory of Complexes．Geometric Consideration 45

2．Elements of the Theory of Complexes．Modulo Two Theory 50

3．The Jordan Curve Theorem 61

4．Proof of the Jordan Curve Theorem 65

5．Some Additional Properties of Complexes 68

6．Closed Surfaces．Generalities 72

7．Closed Surfaces．Reduction to a Normal Form 83

Problems 84

Chapter Ⅲ．Theory of Complexes 86

1．Intuitive Approach 86

2．Simplexes and Simplicial Complexes 87

3．Chains,Cycles,Homology Groups 89

4．Geometric Complexes 95

5．Calculation of the Betti Numbers．The Euler-Poincaré Characteristic 99

6．Relation between Connectedness and Homology 103

7．Circuits 105

Problems 107

Chapter Ⅳ．Transformations of Complexes．Simplicial Approximations and Related Questions 110

1．Set-transformations．Chain-mappings 110

2．Derivation 112

3．The Brouwer Fixed Point Theorem 117

4．Simplicial Approximation 119

5．The Brouwer Degree 124

6．Hopf＇s Classification of Mappings of n-spheres on n-spheres 132

7．Some Theorems on the Sphere 134

Problems 140

Chapter Ⅴ．Further Properties of Homotopy．Fixed Points．Fundamental Group．Homotopy Groups 142

1．Homotopy of Chain-mappings 142

2．Homology in Polyhedra．Relation to Homotopy 148

3．The Lefschetz Fixed Point Theorem for Polyhedra 153

4．The Fundamental Group 157

5．The Homotopy Groups 170

Problems 180

Chapter Ⅵ．Introduction to Manifolds．Duality Theorems 183

1．Differentiable and Other Manifolds 183

2．The Poincare Duality Theorem 188

3．Relative Homology Theory 195

4．Relative Manifolds and Related Duality Theory(Elementary Theory)．Alexander＇s Duality Theorem 202

Problems 206

Bibliography 208

List of Symbols 211

Index 213