Introduction to Topology

Although not described as such, this book is actually a new edition of an existing textbook. The first book has a different title (Elements of Topology), a different publisher (CRC Press) and, somewhat surprisingly, appears to still be in print. Please see the (quite favorable) review of the earlier book for a discussion of its topic coverage and pedagogical features. 

 

The changes in this book appear to be local, rather than global. The tables of contents of the two books are very similar; the most significant change, perhaps, is that the chapter on separation axioms now follows, rather than precedes, the chapters on compactness and topological constructions. Changes in the actual text include the correction of known typographical errors, tweaks in the proofs (on occasion, the author says, some proofs have been completely rewritten), and the inclusion of some items in the text that were formerly left as exercises. 

 

In addition, the Index, about which I registered a mild criticism in the original review, has been enhanced. The author even credits the original review for this change, which is nice of him. In another change from the earlier book, there are now solutions to the exercises. These do not appear in the text itself but in a password-protected solutions manual available on Springer’s website for the book. The solutions manual is itself an impressive body of work: almost 200 pages of text, with solutions to what appears to be every problem in the book (although, and only when appropriate, these solutions sometimes take the form of one-word answers like “trivial”). I suspect that this manual will be quite useful to instructors teaching a course from this book. 

 

The features that I liked about the first edition (the primary one being its broad range of topic coverage, including an introduction to fundamental groups and covering spaces, and a discussion of matrix groups as topological spaces) remain in place here. Also, as in the first edition, there is a good dependence chart, allowing an instructor some flexibility in the use of the text.

 

There is only one respect in which I like the first edition of the book more than this one, and that is a purely personal quibble over which the author presumably had no control: the typography of the book. The font in this text seems considerably smaller than in the first edition; the Index, for example, despite containing probably twice as many entries as the first, comprises the same number of pages. This gives the book a rather dense appearance, compared to what I would describe as the more open and inviting appearance of the CRC book.

 

But this is, as I say, a quibble. Overall, this new edition has made a good book even better. This book would make a good choice for a graduate course in point-set (with an introduction to algebraic) topology, and would also function well as a text for a fairly sophisticated undergraduate course.

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Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.