Customer Reviews

Topology (Dover Books on Mathematics)

5 star:		(5)
4 star:		(1)
3 star:		(1)
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When I was a graduate student 40 years ago there were very few texts in topology. The only two that I recall being in use were Hocking and Young and the book by Kelley. Over the years my copy of Hocking and Young has become quite worn. It is a wonderful book that gives the true flavor of topology. It is also contains a large number of topics that one can refer to later on. It becomes quite apparent very earlier that no one will be able to fully appreciate the book in the time span of one course. It is a book that must be read and reread over and over again. It is a real classic. I do not believe that it is the type of book that would be of much or any general interest but to a point set topologist it is a classic and must for his bookself. I am quite surprised over its low price. I can not help but compare it with the newer book by Munkres. I recall seeing Munkres book many years ago and disliking it. But the current edition seems much closer in flavor to HY and Munkres book is quite good. Munkres style is much clearer than HY, but both books target a very specialized group of people. Neither book is for the faint of heart and will take many years to absorb. Considering that Munkres book is 9 times as expensive as HY, HY seems to be the better buy.

I am teaching myself topology with this book right now, and I must say it has an excellent balance of motivation and rigor. The very first definition in the book reveals the implications of topology to anyone who has studied limit pts (and how connectedness is defined in terms of same). After less than a week of study, I understood the big picture better than most people I know who have taken a full course. The exercises are a little sparse, perhaps, but they generally make up for their small number with increased difficulty. I have only encountered a few exercises that I could call trivial. My only gripe is that the exercises are sometimes a little tricky to find.

Hocking's book on point-set and algebraic topology is better suited for self-study than any other books covering these subjects that I have read. Any student who has taken one or two analysis/algebra courses might give it a try.

I find many areas of this text as lucid and captivating as some of the most famous topology works. While I prefer Munkres' text in a classroom setting, Hocking and Young give an enjoyably different approach with more discussion in some interesting topics. It serves as possibly the best book for self teaching or quick reference.

The book has its virtues, sure enough. But there are some downsides
to it as well that I feel are underrepresented in the other reviews so far.

Let me first note that, contrary to the statement of one other reviewer, there are exercises in this book, and not too few. However, I found that I did not need them, since thinking deeply about all the little flaws and omissions that are scattered through the text allowed me to mature faster than going through these exercises. Needless to say, though, that this type of exercise can be a bit frustrating. I often found myself wondering if it was my lack of maturity that made me struggle, or if the authors actually made their life too simple at various points. Luckily, I found amply evidence for the latter. For example, the reader familiar with homotopy may open the book on page 164 and inspect their proof that the curve given by f(1-x) is the inverse of that given by f(x) in the fundamental group. While this is a true statement of course, their constructed homotopy to prove this is not really continuous, and a slight modification of it could be used as a "proof" that every curve is homotopy equivalent to a constant one. A useful review of the book by a professional can be found at the following URL,

http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183524657

where similar shortcomings are noted. I agree that the latter will probably not slow down an expert who chooses this book as a reference. For beginners, however, they are unnecessary obstacles. I bought this book because I got attracted by the balanced selection of topics ranging from point set topology to algebraic topology. I wanted to learn the latter, but first needed to become proficient in the former. Having now read only the first part of the book devoted to point set topology, I can say that the book did its job, and did it quite well. However, I cannot shake off the feeling that I could have learned the same material in a fraction of the time from a different book. Feeling that I do now have a solid enough background in point set topology, I am considering to not read the second half of the book, and instead learn algebraic topology from a more modern text.

An excellent book, not for those persons unfamiliar with the topic of topolgy; yet, combined with simpler texts this book is a goldmine of topological theorems and their proofs.

Very clearly written, full of examples and counterexamples, making use of pictures but never sacrificing rigor, the authors of this book have given students of topology a superb introduction to the field. Many students have been educated in topology by using this book, and it is sure to remain a classic in the field. It builds a solid understanding of the basic rudiments and intuition behind point-set, geometric, and algebraic topology. There is a lot of material covered in the book, and some very specialized subjects, such as Cech and Vietoris homology and some dimension theory, but with some preserverance and concentration, the entire book can be grasped within reasonable time constraints. Probably the only minus to the book is the lack of exercises. This is a quite serious omission, for the only way to master a subject is to work problems that require careful thought for their solution.

The beginning student of topology should probably read this book with the following mindset: try to think of ways and techniques that you would devise to study the structure of a topological space. Homotopy and homology (in various forms) are the standard techniques for doing this. These strategies have varying degrees of success, but their use in topology now seems to be reaching a saturation limit, even though the explicit calculation of homotopy groups is still a very active area. New techniques and concepts, representing sort of a "large deviation" from the standard ones discussed in this book, will be needed to make further progress in the study of complicated topological spaces. Something more is needed now, that is completely different than homology and homotopy theory, that will make more transparent the properties of these spaces. These new techniques will be somewhat radical from the standpoint of current ones, but they will be more effective from a conceptual (and computational) point of view.