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
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
Further Editions
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
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces: up to $8.2.3$: Definition:Uniformly Convergent Real Sequence -- may be gaps
- Redoing from start:
- Chapter $5$ in progress -- exercises to do:
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $5$: Compact spaces: Exercise $5.10: 2$