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

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



Source work progress
* : up to $8.2.3$: Definition:Uniform Convergence/Real Numbers -- may be gaps
 * Redoing from start:


 * : $1$: Review of some real analysis: $\S 1.1$: Real Numbers