Introduction to Topology

This text explains nontrivial applications of metric space topology to analysis. Covers metric space, point-set topology, and algebraic topology. 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 homomorphisms
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 Noncontractibility of Sn
3 Simplexes and barycentric subdivision
4 Approximation by piecewise linear maps
5 Degrees of maps
BIBLIOGRAPHY
LIST OF NOTATIONS
SOLUTIONS TO SELECTED EXERCISES
INDEX