Metric Spaces

Authors: Mícheál ó Searcóid



The abstract concepts of metric ces are often perceived as difficult. This book offers a unique approach to the subject which gives readers the advantage of a new perspective familiar from the analysis of a real line. Rather than passing quickly from the definition of a metric to the more abstract concepts of convergence and continuity, the author takes the concrete notion of distance as far as possible, illustrating the text with examples and naturally arising questions. Attention to detail at this stage is designed to prepare the reader to understand the more abstract ideas with relative ease.

The book goes on to provide a thorough exposition of all the standard necessary results of the theory and, in addition, includes selected topics not normally found in introductory books, such as: the Tietze Extension Theorem; the Hausdorff metric and its completeness; and the existence of curves of minimum length. Other features include:
• end-of-chapter summaries and numerous exercises to reinforce what has been learnt;
• extensive cross-referencing to help the reader follow arguments;
• a Cumulative Reference Chart, showing the dependencies throughout the book on a section-by-section basis as an aid to course design.

The book is designed for third- and fourth-year undergraduates and beginning graduates. Readers should have some practical knowledge of differential and integral calculus and have completed a first course in real analysis. With its many examples, careful illustrations, and full solutions to selected exercises, this book provides a gentle introduction that is ideal for self-study and an excellent preparation for applications.

Table of contents

Front Matter
Pages i-xix

Metrics
Pages 1-20

Distance
Pages 21-34

Boundary
Pages 35-51

Open, Closed and Dense Subsets
Pages 53-69

Balls
Pages 71-82

Convergence
Pages 83-102

Bounds
Pages 103-124

Continuity
Pages 125-146

Uniform Continuity
Pages 147-163

Completeness
Pages 165-190

Connectedness
Pages 191-204

Compactness
Pages 205-226

Equivalence
Pages 227-244

Back Matter
Pages 245-304