By Theodore W. Gamelin and Robert Everist Greene. The ideas of metric and metric space, which are the subject matter of this chapter, are abstractions of the concept of distance in Euclidean space. These abstractions have turned out to be particularly fundamental and useful in modern mathematics; in fact, the aspects of the Euclidean idea of distance retained in the abstract version are precisely those that are most useful in a wide range of mathematical activities. The determination of this usefulness was historically a matter of experience and experiment. By now, the reader can be assured, the mathematical utility of the metric-space information developed in this chapter entirely justifies its careful study.

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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition. This is a Dover reprint of a text, first published in , covering the basics of point-set and algebraic topology at a fairly sophisticated undergraduate level.

Unlike many Dover reprints, which are unaltered, this one differs from the original in that it contains an appendix, almost 40 pages long, of solutions to most of the exercises.

The text is divided into four chapters, each subdivided into numerous sections. The chapter on metric spaces is quite complete and contains a mini-course in topology in the context of these spaces, with an emphasis on applications to analysis.

Continuity, compactness and connectedness, and product spaces are introduced and discussed, first for metric spaces and then, in the next chapter, in the more general setting of topological spaces. The metric space chapter in the book under review also pursues some ideas beyond that which is usually done in undergraduate texts at this level.

For example, while it is not uncommon for books to mention normed vector spaces as examples of metric spaces, this book takes matters further and goes into topics that are often associated with courses in functional analysis, such as the definition of a Banach space and a statement and proof of the Uniform Boundedness Theorem.

Complete metric spaces are introduced, which is again pretty standard, but here the authors go so far as to state and prove the contraction mapping theorem, and give examples to analysis integral equations, differential equations. This is by no means the only time that quite nontrivial analysis is discussed in the text; Frechet derivatives, for example, are also the subject of a section in the first chapter, and proofs of the implicit and inverse function theorems in the context of Banach spaces are given.

Chapter 3 on homotopy theory discusses the fundamental group, covering spaces, and related ideas. This final chapter strikes me as being a bit too sophisticated for the average undergraduate. The writing throughout is quite elegant but very concise, perhaps too much so for the current generation of undergraduates.

The first two chapters, for example, which probably contain enough material for most of a one-semester course, comprise only about pages. The cofinite topology is mentioned in the exercises. As mentioned earlier, this Dover edition contains solutions at the end of the book; these range from fairly complete to those that are as concise as the text itself e.

That, I think, summarizes my feelings about the book under review as well — a very nicely written, concise account of the material that certainly belongs on the shelf of anybody interested in topology at its current price of about ten dollars on amazon. Mark Hunacek mhunacek iastate. Skip to main content. Search form Search. Login Join Give Shops. Halmos - Lester R. Ford Awards Merten M. Theodore W. Gamelin and Robert Everist Greene. Publication Date:. Number of Pages:. BLL Rating:. General Topology.

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## Introduction to Topology: Second Edition

Introduction to Topology : Second Edition. Theodore W. Gamelin , Robert Everist Greene. One of the most important milestones in mathematics in the twentieth century was the development of topology as an independent field of study and the subsequent systematic application of topological ideas to other fields of mathematics.

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## Introduction to Topology

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