ProofWiki:Books/W.A. Sutherland/Introduction to Metric and Topological Spaces
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W.A. Sutherland: Introduction to Metric and Topological Spaces
Published 1975, Oxford Science Publications.
ISBN 0-19-853161-3
Subject Matter
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
