Customer Reviews

Counterexamples in Topology (Dover Books on Mathematics)

5 star:		(5)
4 star:		(6)
3 star:		(0)
2 star:		(0)
1 star:		(1)

 

 

This book has examples in it that are "missing", so to speak, from many regular topology books. It aims to shore up some of these shortcomings, with examples that the student can see and understand. There are charts and graphs, as well as a detailed explanation. Some "problems" often found in regular topology books are solved. Very few proofs, if any, are given. This is not a book meant to be studied without a regular textbook on topology, only to be used as an overall review of problems and short basic premises of topology. Use this in addition to your regular fare, but keep it close at hand when doing homework or preparing for an exam.
There are fundamentals on Cantor's Theorem, the countability or uncountability of sets, compactness, closed and bounded functions, open sets, continuity, connectedness, etc. All these are basic to topology, and this book does address them, but in a brief way. It then shows a basic overview of topology that helps greatly to understand the different fields of topology.

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.

This is not your typical topology text. In the first part of the book, the authors give a crash course on basic point-set topology. Rather than proving theorems, the emphasis is on defining and explaining concepts, especially as the various concepts relate to each other. The explanations are not always sufficient in themselves for the student's understanding, but that wasn't the book's mission. In the second part, the book provides "Counterexamples": quite a few topologies, both the predictable and the quirky kinds. The topologies given vary considerably in level of difficulty. Thirdly, several pages at the end provide charts showing which properties the book's listed topologies have. The reader can use these charts to find a suitable topology for many applications or disproofs. Counterexamples in Topology is both useful and enjoyable, particularly for people who benefit from charts and outlines. It is not a book to be plowed through, page by page, like a textbook. Rather, it is a compendium of several interesting cases, any of which can be studied independently of the rest.

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.

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.

This book is a must have for anyone who wants to contribute to remaining questions in point-set topology. Property inheritance relationship diagrams fill the book, quickly giving someone a good knowledge of all the classic point-set properties of spaces more thoroughly than is ever taught in grad-school these days. The only draw back is that the book and counterexamples deal strictly with point-set. How nice it would be to have a new edition (or volume two) of this type of book pertaining to algebraic topology. For example, what is an example of a non-contractible space with all zero homotopy groups? This question (and any algebraic top. questions) wont be answered in Steen's book.

I really like the compactness of this book; that is, this book is the finite subcover for any arbitrary cover on the subject of Topology. Seriously, It gives almost all the major definitions which you would ever need, and it provides all kinds of nontrivial examples. Depending on your preference, the notation can sometimes be a little awkward, but it is neither incorrect nor ambiguous; that is, it is still correct and clear, depending on your understanding. Any mathematician ought to have this book in his collection for reference.

This is no more and no less than a catalogue (which may or may not be "complete" as far as it goes) of instances of point-set topology. It can't be used as a text book because you need to know your way around the subject a bit before you can make much sense of it. However, if you read it slowly enough to take in every concept (work taking the discipline to read it, yes, *that* slowly) it gives a rewarding insight into "how it all fits together".

One of the stages of development of a mathematical field of thought consists of organising, categorising and arranging. This is what Steen and Seebach set out to do. In their words (paraphrased), every example of topology is in fact a counterexample of something. In this book are gathered about 150 examples of topologies, all of which have a unique configuration of properties.

However, before you can understand what's being talked about, you need to get to grips with the language. The first forty or so pages take you at breakneck speed through all the topological concepts, from limits, closures, interiors, functions and filters, through separation axioms, compactness and connectedness, to metric spaces. Only when the basics are out of the way do you get onto the examples proper.

Beware: the book is not perfect. There are mistakes. The first one that I found was on page 7:
"The exterior of the union of sets is always contained in the intersection of the exteriors, and similarly, the exterior of the intersection is contained in the union of the exteriors; equality holds only for finite unions and intersections."
Not at all. The union of exteriors is contained in the exterior of the intersections, and equality does not necessarily hold even for the exteriors of two sets.

If you want to use this as a study aid, then I recommend you first work through that first part (only 40 pages) and determine to yourself how you would prove each and every one of the statements made without proof. For there are many - and, as we have seen, there exists at least one instance where the statements are wrong.

Brilliant little book - but for the mistakes I'd give it 5 stars.