Introduction to topology 拓扑学导论

This volume explains nontrivial applications of metric space topology to analysis, clearly establishing their relationship. Also, topics from elementary algebraic topology focus on concrete results with minimal algebraic formalism. Two chapters consider metric space and point-set topology; the other 2 chapters discuss algebraic topological material. Includes exercises, selected answers and 51 illustrations. 1983 edition.

ONE METRIC SPACES
1.Open and closed sets
2.Completeness
3.The real line
4.Products of metric spaces
5.Compactness
6.Continuous functions
7.Normed linear spaces
8.The contraction principle
9.The Frechet derivative
TWO TOPOLOGICAL SPACES
1.Topological spaces
2.Subspaces
3.Continuous functions
4.Base for a topology
5.Separation axioms
6.Compactness
7.Locally compact spaces
8.Connectedness
9.Path connectedness
10.Finite product spaces
11.Set theory and Zorn's lemma
12.Infinite product spaces
13.Quotient spaces
THREE HOMOTOPY THEORY
1.Groups
2.Homotopic paths
3.The fundamental group
4.Induced horIiomorphisms
5.Covering spaces
6.Some applications of the index
7.Homotopic maps
8.Maps into the punctured plane
9.Vector fields
10.The Jordan Curve Theorem
FOUR HIGHER DIMENSIONAL HOMOTOPY
1.Higher homotopy groups
2.Noncontractibflity of Sn
3.Simplexes and barycentric subdivision
4.Approximation by piecewise linear maps
5.Degrees of maps
BIBLIOGRAPHY
SOLUTIONS TO SELECTED EXERCISES
INDEX