Elementary Point-Set Topology: A Transition to Advanced Mathematics

This is a back-to-basics introductory text in point-set topology that can double as a transition to proofs course. The writing is very clear, not too concise or too wordy. The book was developed from many years of lectures at Seattle University and the authors recommend it for a one-semester course, or possibly two semesters, depending on the background and interests of the students.

The treatment covers topological spaces in full generality, but it keeps a concrete focus because most of the examples are taken from the real line or the plane. It is not a topology-for-analysis book but really does concentrate on the topological concerns, for example with a lengthy chapter on connectedness. I thought the approach was a little too pure, because it ignores the motivation from real analysis, but it makes up for some of this purity with the last chapter, that comprises a series of interesting applications all based on fixed-point principles.

Each section of the book ends with a large number of exercises. For the most part these are not very hard and are intended to test your comprehension of the material in the section. The optional first chapter covers set theory and proof methods; if the students already know this material you can start with Chapter 2 to present a straight topology course, otherwise the book can be used as an introduction to proofs course also.

The publisher, Dover, is a famed reprint house that has occasionally published a new work, but it has now started a series of new works under the name “Aurora: Dover Modern Math Originals”. The stated purpose of the series is: “Now, we are proud to present this important new series of all-original texts by some of today's most prominent scholars. Each edition is presented in an affordable paperback format designed to withstand years of classroom use.” In other words, they aim to provide good textbooks at the usual Dover bargain prices ($25 for a complete topology course, in this instance). I think this is a worthy goal and they have accomplished it in this text.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is allenstenger.com. His mathematical interests are number theory and classical analysis.