Metric Spaces

The abstract concepts of metric spaces are often perceived as difficult. This book offers a unique approach to the subject which gives readers the advantage of a new perspective on ideas 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.

The book provides 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.

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.

• Continuity
• Convergence
• Distance
• Metric space theory
• Metric spaces
• Open sets
• calculus
• compactness
• minimum

From the reviews:

"This book is truly about metric spaces. … The book is packed full of material which does not often appear in comparable books. … His use of questions to increase understanding and to move on to the next topic are also to be appreciated. … this is a great book and suitable … for third-and fourth-year under-graduates and beginning graduate students." (Marion Cohen, MathDL, January, 2007)

"The book is very readable. It includes appendixes on the necessary mathematical logic and set theory, and has a substantial number of exercises… Every concept is demonstrated via a large number of examples, starting with commonplace ones and expanding the reader’s horizon with the more abstruse ones, to give a sense of the scope of the concepts… A useful addition to any library supporting an undergraduate mathematics major." (D. Z. Spicer, CHOICE, March, 2007)

• UCD School of Mathematical Sciences, University College Dublin, Belfield Dublin 4, Ireland

  Mícheál Ó Searcóid

Mícheál Ó Searcóid is author of another, higher-level, SUMS book, Elements of Abstract Analysis, 1-85233-424-X, published November 2001, sales (as of June 2005): 1051 (ROW: 634; US: 417).

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