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

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

Published $\text {1975}$, Oxford Science Publications

ISBN 0-19-853161-3.


Subject Matter

Metric Spaces
Topology


Contents

Introduction
Notation and Terminology
$1$. Review of some Real Analysis
$1.1$. Real numbers
$1.2$. Real sequences
$1.3$. Limits of functions
$1.4$. Continuity
$2$. Continuity Generalized: Metric Spaces
$2.1$. Motivation
$2.2$. Examples
$2.3$. Open sets in metric spaces
$2.4$. Equivalent metrics
$2.5$. Continuity
$3$. Continuity Generalized: Topological Spaces
$3.1$. Topological spaces
$3.2$. Bases
$3.3$. Sub-bases and weak topologies
$3.4$. Subspaces
$3.5$. Products
$3.6$. Homeomorphisms
$3.7$. Definitions
$3.8$. Quotient spaces
$4$. The Hausdorff Condition
$4.1$. Motivation
$4.2$. Separation axioms
$5$. Compact Spaces
$5.1$. Motivation
$5.2$. Definition of compactness
$5.3$. Compactness of $\left[{a, b}\right]$
$5.4$. Properties of compact spaces
$5.5$. Continuous maps on compact spaces
$5.6$. Compactness and constructions
$5.7$. Compact subspaces of $\R^n$
$5.8$. Compactness and uniform continuity
$5.9$. An inverse function theorem
$6$. Connected Spaces
$6.1$. Introduction
$6.2$. Connectedness
$6.3$. Path-connectedness
$6.4$. Comparison of definitions
$6.5$. Components
$7$. Compactness Again: Convergence in Metric Spaces
$7.1$. Introduction
$7.2$. Sequential compactness
$8$. Uniform Convergence
$8.1$. Introduction
$8.2$. Definition and examples
$8.3$. Cauchy's criterion
$8.4$. Uniform limits of sequences
$8.5$. Generalizations
$9$. Complete Metric Spaces
$9.1$. Introduction
$9.2$. Definition and examples
$9.3$. Fixed point theorems
$9.4$. The contraction mapping theorem
$9.5$. Cantor's and Baire's theorems
$10$. Criteria for Compactness in Metric Spaces
$10.1$. A general criterion
$10.2$. ArzelĂ -Ascoli Theorem
$11$. Appendix
$11.1$. Real numbers
$11.2$. Completion of metric spaces
$12$. Guide to Exercises
Bibliography
Index


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