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An Introduction to Metric and Topological Spaces (Second Edition)

Wilson A. Sutherland
OXFORD UNIVERSITY PRESS 2009, 224 PAGES PRICE (HARDBACK) £45.00 ISBN 978-0-19-956307-4

Second editions of maths textbooks occupy a strange place in the literary universe. They are not really equivalent to a ‘Greatest Hits’ album, on which only the best examples from a long career survive. Nor can they be considered as being analogous to a Director’s Cut of a movie, in which creativity is given free reign over commercially-dictated constraints on the maximum time of the film.

They can, however, provide an opportunity to update a book’s contents to reflect recent developments in the field. Alternatively, they may allow the presentation of the material to be refreshed to reflect the expectations of a new generation of readers. The Second Edition under review here falls more into the latter category than the former.
The target audience for this work comprises undergraduates who have completed a course on the analysis of real-valued functions of one variable. This background is used to review concepts like sequences, limits of functions and continuity in a familiar context. Having established this firm foundation the author generalises these concepts to metric spaces, talking about open and closed sets, interiors and boundaries. The generalisation is taken further to introduce topological spaces, subspaces and product spaces. Connected, compact and quotient spaces are also covered.

The final topic is an example of one that has been amplified in this Second Edition. This has been made possible by the introduction of a companion website, which is quickly becoming a ‘must have’ addition to any new textbook. This website includes a small amount of material from the First Edition, themovement of which has, for example, provided space for the additional discussion of quotient spaces. The website is also used to provide more detail on tangential topics than would have been possible in a footnote. For example, a pathological example is presented to show the importance of including the Hausdorff condition in the general definition of a closed surface.

A significant strength of this text is the large number of well selected exercises. Even a relatively small seven page chapter on Sequential Compactness is followed by seventeen items to challenge the reader. Unlike the First Edition, answers to the exercises are not included in the book, although they are available on a limited access part of the website. I suspect that an individual’s view on whether this is a change for the better will depend heavily on which side of the lectern they occupy during class.

It is clear that the author has considerable experience in communicating the subject matter, adeptlywarning his readers of possible pitfalls or potential sources of confusion. One example is the potential confusion between the term‘(a,b)’ being interpreted as an interval of the real line and it being interpreted as the location of a point in two-dimensional space. Another example clarifies that the terminology ‘Lipschitz equivalence’ is not universally adopted.
In addition to this clarity, there is also a welcome sprinkling of humour: it is noted that subsets of metric spaces are ‘nothing like doors’ – that is they can be neither open nor closed. Additionally, it is noted that a space is Hausdorff if any two distinct points can be ‘housed off’ from each other by disjoint open sets. Even the old joke about a topologist’s inability to distinguish between a coffee cup and a doughnut gets an airing.

The First Edition of this work was highly praised. No lesser journal than the Bulletin of the IMA observed that it was ‘a well written and to be recommended text’. It is reassuring to note that the Second Edition is equally impressive. The changes that have been made have only served to enhance the book. Hence, it remains a highly recommended introduction to metric and topological spaces.

Rob Ashmore CMath FIMA, CSci
Defence Science and Technology Laboratory
Mathematics Today December 2011

The views and opinions expressed herein are those of the author and do not necessarily reflect those of the Defence Science and Technology Laboratory.

An Introduction to Metric and Topological Spaces can be purchased at Amazon.co.uk