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Topological and uniform spaces. (English) Zbl 0625.54001
Undergraduate Texts in Mathematics. New York etc.: Springer-Verlag. IX, 163 p.; DM 74.00 (1987).
[This book contains the following chapters: topological spaces, continuity, the induced topology and its dual, open functions and closed functions, compact spaces, separation conditions, uniform spaces, the uniform topology, connectedness, countability and related topics, functional separation conditions, completeness and completion.]
After reading this book I felt somewhat dissatisfied. The author presents all the basics: continuity, convergence, separation, compactness etc., but I think that someone trying to learn topology from this book may get the feeling that it is mostly a matter of routine checking. Many statements that are false, are seen to be so via either discrete or indiscrete examples, in fact the only nonseparable metric spaces mentioned as such that I found were discrete. The book could have been much livelier by the introduction of a few more nontrivial examples. The book does have a few good points though: the approach to compactness is very nice and the proof that uniformizable spaces are completely regular is one of the slickest I have ever seen. There are also a few errors in the book but I leave those for the reader to find, sometimes that’s the nicest part of reading a book like this.
Reviewer: K.P.Hart

MSC:
54-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to general topology
54E15 Uniform structures and generalizations