《LECTURE NOTES ON ELEMENTARY TOPOLOGY AND GEOMETRY》

Chapter 1Some point set topology1

1.1 Naive set theory1

1.2 Topological spaces5

1.3 Connected and compact spaces11

1.4 Continuous functions13

1.5 Product spaces16

1.6 The Tychonoff theorem20

Chapter 2More point set topology26

2.1 Separation axioms26

2.2 Separation by continuous functions31

2.3 More separability34

2.4 Complete metric spaces40

2.5 Applications43

Chapter 3Fundamental group and covering spaces49

3.1 Homotopy49

3.2 Fundamental group52

3.3 Covering spaces62

Chapter 4Simplicial complexes78

4.1 Geometry of simplicial complexes79

4.2 Barycentric subdivisions83

4.3 Simplicial approximation theorem90

4.4 Fundamental approup of a simplicial complex94

Chapter 5Manifolds109

5.1 Differentiable manifolds109

5.2 Differential forms118

5.3 Miscellaneous facts132

Chapter 6Homology theory and the De Rham theory153

6.1 Simplicial homology153

6.2 De Rham’s theorem161

Chapter 7Intrinsic Riemannian geometry of surfaces175

7.1 Parallel translation and connections175

7.2 Structural equations and curvature184

7.3 Interpretation of curvature190

7.4 Geodesic coordinate systems198

7.5 Isometries and spaces of constant curvature207

Chapter 8Imbedded manifolds in R3216

Bibliography230

Index231