ENGLISH

Metric Spaces

By Mícheál Ó Searcóid (auth.)

Book information

Publisher
Springer-Verlag London
Year
2007
ISBN
978-1-84628-369-7, 978-1-84628-627-8
Language
english
Format
PDF
Filesize
4 MB (4559410 bytes)
Series
Springer Undergraduate Mathematics Series
Edition
1
Pages
304\317
Time added
2013-08-01 04:00:00
Analysis; Functional Analysis; Topology

Description

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. 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

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