Book:W.A. Sutherland/Introduction to Metric and Topological Spaces

From ProofWiki
Jump to: navigation, search

W.A. Sutherland: Introduction to Metric and Topological Spaces

Published $1975$, Oxford Science Publications

ISBN 0-19-853161-3.


Subject Matter

Metric Spaces
Topology


Contents

Introduction
Notation and Terminology
Review of some Real Analysis
Real numbers
Real sequences
Limits of functions
Continuity
Continuity Generalized: Metric Spaces
Motivation
Examples
Open sets in metric spaces
Equivalent metrics
Continuity
Continuity Generalized: Topological Spaces
Topological spaces
Bases
Sub-bases and weak topologies
Subspaces
Products
Homeomorphisms
Definitions
Quotient spaces
The Hausdorff Condition
Motivation
Separation axioms
Compact Spaces
Motivation
Definition of compactness
Compactness of $\left[{a, b}\right]$
Properties of compact spaces
Continuous maps on compact spaces
Compactness and constructions
Compact subspaces of $\R^n$
Compactness and uniform continuity
An inverse function theorem
Connected Spaces
Introduction
Connectedness
Path-connectedness
Comparison of definitions
Components
Compactness Again: Convergence in Metric Spaces
Introduction
Sequential compactness
Uniform Convergence
Introduction
Definition and examples
Cauchy's criterion
Uniform limits of sequences
Generalizations
Complete Metric Spaces
Introduction
Definition and examples
Fixed point theorems
The contraction mapping theorem
Cantor's and Baire's theorems
Criteria for Compactness in Metric Spaces
A general criterion
Arzelà-Ascoli Theorem
Appendix
Real numbers
Completion of metric spaces