Basic Topology (Undergraduate Texts in Mathematics) [Hardcover]

I'm surprised that several previous reviewers have given this book low ratings. This book is far superior to the standard introductions.

As someone who has studied topology for several years now, I have found that the greatest failing of many introductory texts is the inability to give a real 'feel' for the subject. By 'feel' I mean not only familiarity with the necessary tools and ways of thought needed to progress to higher levels of understanding but also experience with the kinds of problems that plague(excite?) topologists on a daily basis.

Several texts proceed in the logical progression from point set topology to algebraic topology. Munkres is among the best of this style. But the logical order is not always pedagogically best, especially in topology. To start one's topology career by spending one or more semesters on point set topology is utterly ridiculous, given that such point set subtleties are to a large degree not used to study the beginnings of geometric or algebraic topology. This is how these texts fail to give students the 'feel' for topology; the student has no idea what it is that most topologists do, and in fact will not get a good idea until much later.

Armstrong tries (and succeeds for the most part) in grounding concepts in real applications, the way the tools are actually used by research mathematicians. Perhaps this is part of why it may be confusing to the novice; introducing topological groups and group actions on spaces right after the section on quotient spaces may appear a bit much, but those concepts are a big part of *why* quotient spaces are so important! Incidentally, the material on quotient spaces is the most complete I've ever seen in an introductory book; Armstrong covers cones and also gluing/attaching maps.

The book is certainly fun. Imagine learning about space-filling curves right after the section on continuous functions. Armstrong keeps things spiced up throughout the book. He also goes at some length into triangulations, simplicial approximation, and simplicial homology. Then he *applies* this stuff to get results like Borsuk-Ulam, Lefschetz fixed-pt thm, and of course dimension invariance. Throw in less standard material like Seifert surfaces, and you have quite an interesting mix.

The exercises can be quite varied and hard, but are designed to give the reader a realistic view of the difficulties of the subject. The reader will get considerable insight from them, and loads of fun too. I say this, because as someone who already knows the stuff, I find more than a few of the problems enjoyable even now.

Having wrote all that, I should add that I did *not* learn out of this book! But I wish greatly that I had! I would have known sooner whether topology was the right subject for me to pursue and had some 'lead time' to absorb some very fundamental concepts early on. If you pass over this book, be warned that you are shorting yourself in the long run.

Many of the standard introductions to Topology (Munkres comes to mind) focus more on the logical flow of the material, and less on the motivation for the material. This book focuses on the motivation, but after the first few chapters, the logical development is sound too.

The Armstrong book starts out with some fairly advanced concepts, outlining some interesting topological results before giving the modern definition of topological spaces in terms of open sets. Typically, authors give the open set definition of a Topology at the outset, before explaining what topology really is, and without explaining why that definition is used or how it was developed. Armstrong instead shows the historical motivation of the subject, and actually leads the reader through this development, starting with the less elegant but more intuitive definition of spaces in terms of neighborhoods. The equivalent open set definition is then taken in chapter two. However, once things get going, this book does not move slowly at all--quotient spaces and the fundamental group are presented early and covered in depth, and it is not long before the reader encounters genuinely advanced material, in rigorous form.

It's true that this book doesn't cover the same amount of raw material that a book like the Munkres does, and it's true that the book does not follow the most concise logical order, but it offers history, motivation, and initial exposure to more interesting results. Perhaps more importantly, it develops the reader's intuition. In many ways, this book is a complement to the Munkres, and an enthusiastic self-learner would benefit greatly from using both books simultaneously.

At the same time, this book does get into some more advanced topics. It has a particularly clear exposition of simplicial homology. My last word of praise about this book is that although it gives lots of motivation, it is still very concise. I think it's hard to go wrong with this book.

I can see how this book has left very few people happy. To generalize broadly, one often finds two types of reviewers of math books on Amazon: those who find the text too difficult, and denounce it; and those who implicitly denounce the first group by means of vigorous support of the book in question. Typically this latter group goes on to write reviews of books *supported* by the first group in which they denounce the excessive "hand-holding," the pandering to the reader's "intuition," and the general attempts to make the material accessible.

This book, however, manages to both require a non-trivial amount of effort and sophistication from the reader (thus alienating the first group), all while also appealing to intuition and giving large numbers of examples (thus alienating the second).

The following example should make the author's approach clear. On several occasions, Armstrong gives a non-standard definition of an idea. This is usually a definition that is more intuitive (to the beginner), but which is harder to use to complete proofs. This non-standard definition is followed by the standard definition, and the equivalence of both formulations is established. This is the case with connectedness, for instance. First, connectedness is defined by appealing to the idea that a space "should be one piece," leading to the formulation that whenever a connected topological space is decomposed into two subsets, the intersection of the closure of one of these sets with the other set is always nonempty. Soon thereafter, the standard formulation (the formulation which one almost always uses to actually write proofs) is introduced and established as equivalent, namely that a connected topological space is one in which the only sets that are both open and closed are the entire space itself and the empty set.

It is true that this approach makes for a bad reference book. It is also certainly not the most elegant and streamlined presentation. But the book is clearly not meant to be a reference or to be a showcase of exceptional concision and elegance. It is meant to be a book to learn from. Adding to this, the chapters are all full of examples, many of them quite interesting.

I will concede that the writing and layout can be irritating at times. In particular, as has been pointed out many times before, the author does not isolate and highlight all definitions and corollaries. So this adds to the difficulty of using the book as a reference, and it even makes it somewhat unpleasant to read as a learning text at times. But to say that the author does not define things is simply wrong. (In the first chapter the author sketches an overview of the material contained in the text, and it consequently does not contain many formal definitions or proofs. By and large, however, all subsequent chapters are independent of this chapter. So if you are truly scandalized by someone attempting to give a loose overview of the subject, you are entirely free to skip this chapter and refer to it as necessary (which will be infrequently).)

All and all, I thought this was a good first topology text. You are always given good examples to chew on while you are sorting out the technicalities. The problems are also generally good. While many are fairly straightforward, I have found that they are almost all at least thought provoking, and some develop new material entirely. And there are more than a handful of difficult ones.

Finally, it should be emphasized that one can realistically be introduced to the rudiments of wide range of topics in a single semester: general topology, identification spaces, topological groups, the fundamental group, triangulations (including Seifert-Van Kampen), and simplicial homology. (To be clear, the book contains more than that, but I am only outlining what could be done in about 13 weeks.) Moreover, unlike some texts which are only meant to give the flavor of a subject to undergraduates, I have found that the foundations set by this books were substantial enough to build on.