Counterexamples in Topology

To paraphrase Chandrasekhar's review of Watson's Bessel functions text, this is "a veritable mine of information... indispensable to those who have occasion to use point-set topology." I don't think this book is intended to be a text ( & I think the authors say so), in which case it would be terrible because it doesn't explain the concepts very much. It's mostly a catalogue of every kind of set you can come up with, every kind of topology you can put on it, and what properties it has such as what T_i axioms the space satisfies, whether it's compact, para compact, etc etc. Most of the time such things are proven, but be prepared to think hard sometimes about the proofs or fill in details. I'm the kind of student where I have trouble understanding things which are highly 'counter-intuitive' so I had trouble proving things, even when I knew definitions, when I did topology for the first time last term. Once I saw this book though I got used to all the weird things in topology (like the ordered square, R in the lower-limit topology, Sorgenfrey plane, etc etc). This book is incredibly useful as a reference.

A distinct characteristic of point set topology is that it builds on counterexamples. If you thumb through any PST text, many theorems are in the form "If the space T is A,B,C, then the space is X,Y,Z". The point of point set topology (pun unintended) is too determine what A,B,C are, and to weaken the hypothesis. "Can we take condition B out? Maybe hypothesis C can be weaken considerably?" How can we answer these questions? You're right, by counterexamples. Students who want to master point set topology should know the various counterexamples, no matter how contrived or unnatural they seem. While textbooks usually present a counterexample to show why Theorem Three Point Five Oh will not work on a weaker assumption -- most students (and teachers) tend to skip these parts. A collection of counterexamples presented in this book (excellent organisation, by the way) is an essential supplement of a topology course; it enables one to 'see' between the points, so to speak.

As a graduate I encountered a book called "counter examples in analysis" which I found very useful. I always dreamed of such a book in topology, this book exceeds my dreams. It is great. It does not cover all the examples that I have used over the decades but it does cover some that I have never seen. The style is quite readable for a professional topologist. The book goes into a lot of interesting details (and some while not interesting to me would be another person). In short for me it is an essential book. The question is to whom else would this be interesting to. It is clearly of little use to a first year student and less to more advanced student. It's brand of topology is not the current cutting edge. So the audience for this book is limited to a small group and for these people it is top notch.

This is an excellent book to really start understanding all the general topology learned in an introductory (undergrad or grad level) class. The first section of the book is basically a terminology review. The second part of the book is the real meat here and contains all the counter-examples. These spaces tend to clarify all the concepts, their differences and relative strengths and weaknesses. Of course the nice introduction to meterization theory in the appendix also adds value to the book. In short no student of topology should be without this book.

On its own, this is not a good book to learn topology from. When combined with a standard textbook in topology (such as Baum's book) it makes an invaluable guide for the student. For the mathematician, this is an excellent handbook.