Customer Reviews

General Topology (Dover Books on Mathematics)

5 star:		(10)
4 star:		(1)
3 star:		(1)
2 star:		(1)
1 star:		(0)

 

 

One of the purest and most intellectually challenging branches of modern mathematics, general topology is not a subject for the faint hearted. So it was a pleasure when I first encountered one of the best reference introductions to the subject to have seen the light of day. Willard's book remains one of my all-time favourites. It covers everything the aspiring topologist needs to know, and certainly supplies more than enough information for a potential PhD student to choose their initial area of specialisation. The chapters are split intelligently into sub-topics which move at a sensible pace from its introductory notes on essential set theory, through subspaces, products, compactness, separation and countability axioms, compactifications, and function spaces. Many of the "standard spaces" of general topology are introduced and examined in the large number of related problems accompanying each section. And for those wanting a bit more context than a maths book normally provides there's a detailed collection of historical notes for each chapter.

First a caveate: This book may not be the most suitable for everyone that takes a FIRST course on General Topology unless he or she is prepared to put in quite a lot of work. This is because the book contains so much information in relatively few pages that the material is necessarily quite dense. Even so the book is a good purchase because it's cheap and will serve everyone good later as a reference.

The organization of the book: Everything is presented in a perfectly logical order, beginning with a summary of Set Theory and ending with topologies on Function Spaces. During the course the reader is invited to make excursions to other areas of mathematics from a topological point of view and perhaps gain insights into those fields that even specialists don't have. This is mostly done through problems for the reader to solve.

Definitions and Theorems: The definitions are always the most general possible, often presented as a set of axioms that the defined quantity has to fulfill. The theorems are almost always presented in their most general form.

The Proofs: The proofs are generally on either the shortest and most elegant form possible, or taken from the original publications. This is for the benefit of the reader even though it might appear to some readers as "terse" proofs because this kind of proofs is the one that gains the reader the most insight once they are understood. "Short and elegant" does NOT mean that the author leaves out details (unless they are explicitely assigned as problems).

Explanations and Motivations: The text is short and to the point. This again does not mean that the author leaves out anything relevant or that he does not warn for possible pitfalls.

Examples of introduced concepts and definitions: There are numerous well chosen examples, often nontrivial, to illustrate the meaning of introduced concepts.

The problem set: The set of problems is just fantastic. The problems are numerous, diverse, illustrative, and again, sometimes HIGHLY nontrivial. Don't be too scared though, because the author provides very accurate hints of how to approach the more difficult ones.

Bibliography, Historical Remarks and Index: One just has to admire the amount of work the author has put into this.

Miscellaneous: As mentioned, the material is (necessarily) condensed, but the text is never "dry" or boring. There is an undertone of humour in quite a few places. For instance, when the author mentions that not every regular space is completely regular, because there exists a formidable example that shows this fact, he relegates that example to problem 18G "where most people won't be bothered with it". This practically guarantees that most people WILL be bothered by it by looking up 18.G. There, in 18G, he provides som many hints that it is actually doable for most people to reconstruct this formidable (i.e. difficult) example.

On the Downside: There are no solved problems, and the author does not teach the reader on HOW to solve problems. This is however compensated for by the numerous hints in the problem set and through the methods of thaught one learns from reading and understanding the proofs. Also, in topology, one basically has to invent ones own mothod to solve an unsolved problem. There is no canonical way of doing things!

I have yet to see Dugundji's topology text (it's always checked out at my university library) but I would still guess that this is one of the best there is. Willard's book most certainly covers all the topics that "every young analyst should know", as Kelley wrote, but I think this book outdoes Kelley's when it comes to that! It covers topics like convergence, separation & countability, compactness, connectedness, uniform spaces & a short discussion on function spaces & C* algebras at the end. Many of the theorems, proofs & examples come directly from the original source articles, or are the most general versions there is. Each section has a very detailed & interesting historical discussion at the end of the book where the author lists the original articles & the circumstances in which they were published & other stuff. Where the text really stands out though is the problem sets. As the reader goes deeper into the book of course the concepts get more complicated & proofs of extremely deep & important results are outlined as problems. These are things such as the Cantor-Bernstein theorem, Hahn-Banach theorem, Pontryagin duality theorem, stuff about realcompactness, Edwin Hewitt's construction of a regular T1 space in which every continuous function is constant(!) & many more. Don't be afraid though; the discussion in the text, hints given & notes in the back help with proving things like those. Many of the examples are also highly nontrivial & therefore very helpful (imho). To sum up, I believe this book is very underrated & deserves the recognition of the texts by Munkres & Kelley.

After looking at several books on topology, I would have to say that Wilard's General Topology is an excellent resource book. For those who have taken a topology course and want a little more practice with problems, this book has numerious exercises that help form an solid knowledge base. What else is nice about the problems is they are good research-starters for undergraduates. The examples in the chapters are non-trivial and explain the ideas of the chapter. Also, Wilard's General Topology has a slight set-theoritic view to topology, so those who like set-theory and topology, this book will be of great use. I suggest Wilard's General Topology if you need another topology book to help explain ideas from class or other books.

Willard's text is a great introduction to the subject, suitable for use in a graduate course. I am personally not training to be a topologist but I must say that I enjoyed this book thoroughly and walked away with a firmer appreciation of the subject than I had previously had.

There is quite a bit of content ranging from subject matter and an extensive bibliography to a collection of historical notes. The exercises are suitable and doable; I have personally found that most of them range from being easy to moderately challenging but there are plenty of difficult problems as well.

It is important to note, however, that this text is primarily focused on point-set topology. There is a brief exposition of homotopy theory and the fundamental group but nothing compared to, say Munkres. But this is by no means a drawback. Willard thoroughly examines many topics that Munkres sometimes allocates to the exercises. A good example of this is net convergence, a topic that in my opinion, ought to be treated in any introductory topology course. In fact, Willard's development of nets makes for a nice, quick proof of the Tychonoff Theorem while Munkres's approach necessitates the development of a few technical lemmas.

Overall, this book is quite pleasant to read. It is also quite pleasant to purchase compared to several other introductory texts that run anywhere from 50.00-100.00. There are many nontrivial aspects to topology and this book has a way of gently nudging the reader into some of the more technical and delicate aspects of the theory. But as I mentioned before, while this book is a great introduction to point-set topology, this is not the text to read if one is searching for an introduction to algebraic or differential topology. In the latter case, Munkres or Fulton would be a good bet.

This is certainly one of the best books on general topology available. It requires more maturity from the reader than the usual Munkres/Armstrong standard, but IMHO it is perfectly adequate for a first contact with the subject. It is a dense book, and it does not talk much like other books, but the exposition is so clear that this is actually a quality. Being succint, it manages to cover a lot more ground than the standard references; there is much more here than a one-semester course can cover. The exercises are usually difficult; some of them are real challenges (e.g. can you find an order in which the real numbers are well-ordered? This question pops out in the first set of exercises). The exercises are actually the purpose why this book leaves its rivals far behind. They provide the reader with a deep topological way of thinking in many ways: by forcing the reader to construct counterexamples himself (an essential skill for a topologist) and generalizing the theorems presented in the text, often to explore a new technique or construction. Sometimes this may provide the reader with multiple ways to look at a particular problem, which is certainly an useful skill (not to say inspiring!). A good example is the way the author explores the interconnection between nets and filters, which provide two different frameworks for describing topologies by means of convergence. Most other books describe just one approach or the other, and even when they do both they seldom explicit how they are related. A careful reader who works throughout the whole text, or at least through most of it, will have a better understanding of topology than the reader of the more usual texts. For the sake of comparison, I should say I found the discussion here about quotient spaces far clearer than Munkres's. Willard makes clear from the beggining the distinction between the "quotient approach" and the more intuitive "identification approach", which is the formalization of the intuitive grasp of cutting and pasting spaces. The author carefully develops both points of view, to show in the end they are really the same (in the sense of an universal property - i.e., up to homeomorphism). It becomes absolutely clear then that the first, more abstract approach, gives an effective way for manipulating mathematically problems arising in the second, hence its not-so-obvious-at-a-first-glance importance.

Readers who are already familiar with the methods and results of general topology and basic algebraic topology will also benefit from this book, specially from the exercises. This, together with "Counterexamples in Topology", by Steen and Seebach, form the best duo for studying general topology for real; this is the best option available for the ambitious student and the aspiring topologist. Also, as they are both Dover, the prices are ridiculously low. For a couple of bucks you may have access to some of the most beautiful treasures of mathematics.

Great book with a lot of good stuff, especially in the exercises. However, if you're not willing to work through the relatively dense exercises, you might want to look for something that reads more easily and come back to this book later.

I'll write this rate for those who doesn't know ANYTHING about topology, as it was my case before meeting this book. As you probably heard, topology is composed of two pillars; the general part and the algebraic one. This book serves as a total reference for the first topic and just an introduction to the other, covering only homotopy and the fundamental group (there are plenty of books that focuses on this).

When I took the first course in topology (I'm not a math student, just a economics student), I spend a lot of time searching for suitable books, such as Dugundji "Topology", James "Topological and Uniform Spaces" [wich is very good also!], Munkres Topology (I find it too basic) and many others. With a great number on books in my desk, I pick the tiny Willard's book and started reading. Then the topology universe showed with no secrets.

The writing is superb and verbose enough to comprehend the whole picture, and proofs doesn't end in an open fashion as in others books in the subject. However, the really gold mine is in the exercises where you really are asked to reconstruct classic examples (and of course counter-examples), but with enough guide to avoid any extreme confusion (but there are exercises very tough!).

As a textbook it's very high recommended for self study if you are motivated enough, but try to have someone around to ask questions that will inevitable be around you.

I taught myself from this book my senior year in high school and I absolutely loved it.

Pros

The cost is amazing.

After a solid introduction to Topology (I read Munkres for this which seems to be the standard), this book gives a natural continuation on the subject. Willard really does motivate the subject and gives very detailed proofs. He covers the subject very well, provides excellent exercises and organizes the chapters very well.

Cons

There are a few errors, for example, in the way subbases are defined, but this is of little consequence.

this is an amazing book. very wisely constructed with a lot of real content.
if i may ask for something more i would ask for an updated version, and solutions for problems.