基本拓扑和几何讲义

Chapter Some point set topology

1.1 Naive set theory

1.2 Topological spaces

1.3 Connected and compact spaces

1.4 Continuous functions

1.5 Product spaces

1.6 The Tychonoff theorem

Chapter 2 More point set topology

2.1 Separation axioms

2.2 Separation by continuous functions

2.3 More separability

2.4 Complete metric spaces

2.5 Applications

Chapter 3 Fundamental group and covering spaces

3.1 Homotopy

3.2 Fundamental group

3.3 Covering spaces

Chapter 4 Simplicial complexes

4.1 Geometry of simplicial complexes

4.2 Baryccntric subdivisions

4.3 Simplicial approximation theorem

4.4 Fundamental group of a simplicial complex

Chapter 5 Manifolds

5.1 Differentiable manifolds

5.2 Differential forms

5.3 Miscellaneous facts

Chapter 6 Homology theory and the De Rham theory

6.1 Simplicial homology

6.2 Do Rhams theorem

Chapter 7 Intrinsic Riemannian geometry of surfaces

7.1 Parallel translation and connections

7.2 Structural equations and curvature

7.3 Interpretation of curvature

7.4 Geodesic coordinate systems

7.5 Isometrics and spaces of constant curvature

Chapter 8 Imbedded manifolds in Ra

Bibliography

Index

At the present time， the average undergraduate mathematics major findsmathematics heavily compartmentalized. After the calculus， he takes a coursein analysis and a course in algebra. Depending upon his interests （or those ofhis department）， he takes courses in special topics. If he is exposed to topology，it is usually straightforward point set topology; if he is exposed to geometry， it is usually classical differential geometry.

The exciting revelations thatthere is some unity in mathematics， that fields overlap， that techniques of onefield have applications in another， are denied the undergraduate. He mustwait until he is well into graduate work to see interconnections， presumablybecause earlier he doesnt know enough.These notes are an attempt to break up this compartmentalization， at leastin topologygeometry. What the student has learned in algebra and advancedcalculus are used to prove some fairly deep results relating geometry， topology， and group theory.