Customer Reviews

General Topology (Graduate Texts in Mathematics)

5 star:		(4)
4 star:		(1)
3 star:		(0)
2 star:		(1)
1 star:		(0)

 

 

John Kelley wanted the title to be "What every young analyst should know", but was convinced (by Halmos, among others) not to use it. Still, it is a very good description of the book. Barry Simon calls it "superb" and recommends that you read it by trying to do the exercises, recurring to the text as needed. But then you would perhaps not pay attention to how wonderful the text is. I believe this is the best-written modern mathematical text. The proofs are clean and extremely elegant. The prose itself is beautiful and frequently witty. Treats topological and uniform spaces at depth and in detail, so as to be both a textbook and a reference. Excels in both capacities. This is mathematics close to poetry.

I was motivated to read this book while in grad school, becasue I needed to understand the French literature in my field (probability). One particular concern is the metrizability of a general topological space. I would say Kelley's book has a spendid presentation on this subject.

Other things in this book are also practically useful. Convergence in the general sense (net or filter) is useful in mathematical finance. The part on locally compactness and paracompactness is a must for anyone working in differential geometry. And if you work in analysis, then the chapter on space of continuous functions is a good reference to look up.

The exercise problems are also good resources when you need some help. I still remember one cute problem on the neighbourhood systems. It helped me understand how a family of seminorms would yield a topology on a linear space.

Evetually, I read this book from cover to cover. And I would say this is one of the best education I've ever received.

If there has to be a complain, the proofs are somewhat hard to read. But this is more or less determined by the nature of the subjects. And when you are well-motivated and equipped with certain mathematical maturity, this problem will gradually go off.

In summary, this book is comprehensive, useful and beautifully written. It is a treasure that every mathematician's library should have.

This is a great book. The proofs are clearly presented, and generally it is easy to understand the motivation behind definitions and theorems. Exercises are relevant, interesting, and well designed, often allowing the reader to discover things that other texts describe in dull detail. Unfortunately, a few exercises (such as "Integration Theory: Junior Grade") seem to pop out of nowhere. I consider this a minor defect. A much larger annoyance is that Kelley defines partial and linear orders in an utterly non-standard and somewhat clumsy way, which ends up affecting a large number of exercises. If you already know something about orderings, you will encounter many surprises; if you know nothing about them, you may get the wrong idea.

I don't hesitate to give this book 5 stars. It is solid! Many reviewers allow too much personal judgement to cloud their appraisal of a certain book. To me I believe it is important to be as dispassionate as possible so that a prospective buyer can make an unbiased decision. Rather than label a book as "bad" or "good" one should focus on some factors such as:
(1) Content: a summary of the main point covered by the book (this is optional). In the case of this book, this is obvious from the title.
(2) the author's approach: Kelly took what I call the "analyst's approach" to topology. This is fine for those who love analysis but don't really care for topology for it's own sake (like me!) By using this approach, those like me are much more inclined to find topology motivating because ones sees it as abstractions of what one is familiar with
(3) the presentation: Kelly gave a simple but "sophisticated" presentation. You will not describe him as very expository but the presentation is excellent. Some people seem to prefer this style and some don't. No, this has nothing to do with the so-called "mathematical maturity" (how do you define that by the way?) What the author expects you to know to understand the book - that is, the intended audience - is usually stated clearly in the preface

I cannot agree with the other reviewers on this. Back in the days when there were hardly any general texts on topology this may have been good. Nowadays there are at least a dozen such that are far better than this. The printing fonts and layout are spidery and primitive and not easy on the eye. The style is rather formal and dry for a subject as rich as this and little effort is put into illustrating the material with background, diagrams or examples. As I said before there is no shortage of better texts amongst which Hocking & Young is worth special mention.

If you study metric spaces, you can analyze the notion of convergence with the help of sequences. But if you study in arbitrary topological spaces, sequences are not enough. You have to study a net, a generalaization of a sequence. So if you want to learn topology deeply, I do recommend this book. Of course, after you master some text books in topology.