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

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

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

ISBN 0-19-853161-3.

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 $\sqbrk {a, b}$
$5.4$. 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
$10.2$. Arzelà-Ascoli Theorem
$11$. Appendix
$11.1$. Real numbers
$11.2$. Completion of metric spaces
$12$. Guide to Exercises
Bibliography
Index

Next

## Errata

### Example of Preimage of Subset under Mapping: $\map g {x, y} = \tuple {x^2 + y^2, x y}$

$2$: Continuity generalized: metric spaces: $2.3$: Open sets in metric spaces: Example $2.3.16 \ \text {(b)}$:

Let $g: \R^2 \to \R^2$ be given by $\map g {x, y} = \tuple {x^2 + y^2, x y}$. Then for example
 $\ds \qquad \ \$ $\ds S$ $=$ $\ds g^{-1} \set {\tuple {0, 2}, \tuple {0, 1} }$ $\ds$ $=$ $\ds \set {\tuple {x, y} \in \R^2: 0 < x^2 + y^2 < 2 \text { and } 0 < x y < 1}$
We know from (a) that $\set {\tuple {x, y} \in \R^2: 0 < x y < 1}$ is the shaded region in Fig. 2.2. Also, $\set {\tuple {x, y} \in \R^2: 0 < x^2 + y^2 < 2}$ is the interior of a disc of radius $2$ with its centre removed. Hence $S$, the set of all $\tuple {x, y}$ satisfying both conditions, is the intersection of the shaded region with the outlined disc, not including any of the boundary curves or lines.

### Letter $\mathsf L$ and Letter $\mathsf T$ are not Homeomorphic

$3$: Continuity generalized: topological spaces: $3.6$: Homeomorphisms: Examples $3.6.2 \ \text{(d)}$:

The letter $\mathsf L$ and the letter $\mathsf T$ (defined as in $\text{(c)}$) are not homeomorphic.

## Source work progress

Redoing from start:
Chapter $5$ in progress -- exercises to do: