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Chapter 1 Some point set topology 1

1.1 Naive set theory 1

1.2 Topological spaces 5

1.3 Connected and compact spaces 11

1.4 Continuous functions 13

1.5 Product spaces 16

1.6 The Tychonoff theorem 20

Chapter 2 More point set topology 26

2.1 Separation axioms 26

2.2 Separation by continuous functions 31

2.3 More separability 34

2.4 Complete metric spaces 40

2.5 Applications 43

Chapter 3 Fundamental group and covering spaces 49

3.1 Homotopy 49

3.2 Fundamental group 52

3.3 Covering spaces 62

Chapter 4 Simplicial complexes 78

4.1 Geometry of simplicial complexes 79

4.2 Barycentric subdivisions 83

4.3 Simplicial approximation theorem 90

4.4 Fundamental approup of a simplicial complex 94

Chapter 5 Manifolds 109

5.1 Differentiable manifolds 109

5.2 Differential forms 118

5.3 Miscellaneous facts 132

Chapter 6 Homology theory and the De Rham theory 153

6.1 Simplicial homology 153

6.2 De Rham’s theorem 161

Chapter 7 Intrinsic Riemannian geometry of surfaces 175

7.1 Parallel translation and connections 175

7.2 Structural equations and curvature 184

7.3 Interpretation of curvature 190

7.4 Geodesic coordinate systems 198

7.5 Isometries and spaces of constant curvature 207

Chapter 8 Imbedded manifolds in R3 216

Bibliography 230

Index 231