拓扑学教程：理论及应用

1 Set and Map

1.1 Set

1.2 Map

1.3 Counting

1.4 Equivalence Relation and Quotient

2.Metric Space

2.1 Metric

2.2 Ball

2.3 Open Subset

2.4 Continuity

2.5 Limit Point

2.6 Closed Subset

3.Graph and Network

3.1 Seven Bridges in KSnigsberg

3.2 Proof of One-Trip Criterion

3.3 Euler Formula

3.4 Application of Euler Formula

4 Topology

4.1 Topological Basis and Subbasis

4.2 Open Subset

4.3 Topological Space

4.4 Comparing Topologies

4.5 Limit Point and Closed Subset

4.6 Closure

5 Basic Topological Concepts

5.1 Continuity

5.2 Homeomorphism

5.3 Subspace

5.4 Product

5.5 Quotient

6. Complex

6.1 Simplicial Complex

6.2 CW-Complex

6.3 Projective Space

6.4 Euler Number

7 Topological Properties

7.1 Hausdorff Space

7.2 Connected Space

7.3 Path Connected Space

7.4 Connected Component

7.5 Compact Space

7.6 Limit Point Compact Space

8 Surface

8.1 Manifold

8.2 Surface

8.3 Simplicial Surface

8.4 Planar Diagram

8.5 Cut and Paste

8.6 Classification of Surface

8.7 Recognition of Surface

9 Topics in Point Set Topology

9.1 Normal Space

9.2 Paracompact Space

9.3 Complete Metric Space

9.4 Baire Category Theorem

9.5 Infinite Product

9.6 Space-Filling Curve

9.7 Space of Maps

Index

The aim of the book is to give a broad introduction to topology for undergraduate students. It covers the most important and useful parts of point-set as well as combinatorial topology. The development of the materia is from simple to complex，concrete to abstract， and appeals to the intuition of the reader. Attention is also paid to how topology is actually used in the other fields of mathematics. Qver 150 illustrations， 160 examples and 600 exercises will help readers to practice and fully understand the subject.