Introduction to Metric and Topological SpacesOne of the ways in which topology has influenced other branches of mathematics in the past few decades is by putting the study of continuity and convergence into a general setting. This new edition of Wilson Sutherland's classic text introduces metric and topological spaces by describing some of that influence. The aim is to move gradually from familiar real analysis to abstract topological spaces, using metric spaces as a bridge between the two. The language of metric and topological spaces is established with continuity as the motivating concept. Several concepts are introduced, first in metric spaces and then repeated for topological spaces, to help convey familiarity. The discussion develops to cover connectedness, compactness and completeness, a trio widely used in the rest of mathematics. Topology also has a more geometric aspect which is familiar in popular expositions of the subject as `rubber-sheet geometry', with pictures of Möbius bands, doughnuts, Klein bottles and the like; this geometric aspect is illustrated by describing some standard surfaces, and it is shown how all this fits into the same story as the more analytic developments. The book is primarily aimed at second- or third-year mathematics students. There are numerous exercises, many of the more challenging ones accompanied by hints, as well as a companion website, with further explanations and examples as well as material supplementary to that in the book. |
Contents
| 1 | |
2 Notation and terminology | 5 |
3 More on sets and functions | 9 |
4 Review of some real analysis | 17 |
5 Metric spaces | 37 |
6 More concepts in metric spaces | 61 |
7 Topological spaces | 77 |
8 Continuity in topological spaces bases | 83 |
11 The Hausdorff condition | 109 |
12 Connected spaces | 113 |
13 Compact spaces | 125 |
14 Sequential compactness | 141 |
15 Quotient spaces and surfaces | 151 |
16 Uniform convergence | 173 |
17 Complete metric spaces | 183 |
| 201 | |
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Common terms and phrases
analysis apply boundary bounded called chapter closed closure complete condition connected consider constant contains continuous function continuous map contradiction converges Corollary cover deduce defined Definition discrete easy equation equivalence relation Euclidean Example Exercise exists f is continuous Figure finite fn(x follows function f given gives graph Hausdorff Hence holds homeomorphic idea induced infinite integer interior intersection interval intuitive inverse later least limit Lipschitz map f means metric space Möbius band non-empty notation open sets particular plane positive Proof Proposition Prove quotient real numbers real-valued functions result satisfies sense sequence sequentially compact Similarly space X subsequence subset subspace Suppose that f surface Theorem topological space true uniformly union unique upper bound