拓扑学

Introduction

1.what is point-set topology about?

2.origin and beginnings

Chapter Ⅰ fundamental concepts

1.the concept of a topological space

2.metric spaces

3.subspaces, disjoint unions and products

4.rases and subbases

5.continuous maps

6.connectedness

7.the hausdorff separation axiom

8.compactness

Chapter Ⅱ topological vector spaces

1.the notion of a topological vector space

2.finite-dimensional vector spaces

3.hilbert spaces

4.banach spaces

5.frechet spaces

6.locally convex topological vector spaces

7.a couple of examples

Chapter Ⅲ the quotient topology

1.the notion of a quotient space

2.quotients and maps

3.properties of quotient spaces

4.examples: homogeneous spaces

5.examples: orbit spaces

6.examples: collapsing a subspace to a point

7.examples: gluing topological spaces together

Chapter Ⅳ completion of metric spaces

1.the completion of a metric space

2.completion of a map

3.completion of normed spaces

Chapter Ⅴ homotopy

1.homotopic maps

2.homotopy equivalence

3.examples

4.categories

5.functors

6.what is algebraic topology?

7.homotopy--what for?

Chapter Ⅵ the two countability axioms

1.first and second countability axioms

2.infinite products

3.the role of the countability axioms

Chapter Ⅶ cw-complexes

1.simplicial complexes

2.cell decompositions

3.the notion of a cw-complex

4.subcomplexes

5.cell attaching

6.why cw-complexes are more flexible

7.yes, but...?

Chapter Ⅷ construction of continuous functions on topological spaces

1.the urysohn lemma

2.the proof of the urysohn lemma

3.the tietze extension lemma

4.partitions of unity and vector bundle sections

5.paracompactness

Chapter Ⅸ covering spaces

1.topological spaces over x

2.the concept of a covering space

3.path lifting

4.introduction to the classification of covering spaces

5.fundamental group and lifting behavior

6.the classification of covering spaces

7.covering transformations and universal cover

8.the role of covering spaces in mathematics

Chapter Ⅹ the theorem of tychonoff

1.an unlikely theorem?

2.what is it good for?

3.the proof

last Chapter

set theory （by theodor br6cker）

references

table of symbols

index

This volume covers approximately the amount of point-set topology that a student who does not intend to specialize in the field should nevertheless know.This is not a whole lot， and in condensed form would occupy perhaps only a small booklet. Our aim， however， was not economy of words， but a lively presentation of the ideas involved， an appeal to intuition in both the immediate and the higher meanings.