Basic Topology

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'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.

I took my first class in topology when I was a sophomore. At that point I had very little background on abstract math. To better grasp this subject I tried several books and notes such as Munkers' "Topology", a book from MIR, and the notes at U. Washington homepage (now in book format, "topological manifolds" by John Lee). All of these books and notes were helpful but none of them had an introduction as good as Armstrongs' "Basic Topology". Munkers, for example, go straight to the definition of a topology on a set (not a common sense concept), whereas Armstrong builds this notion from the palpable properties of the Euclidean space. Furthermore, Armstrong's book presents a good variety of topics which enables the reader to cover algebraic topology without spending an entire semester in point set topology. A nice companion for this book, in a perhaps a bit more advanced tone, is Hatcher's "Algebraic Topology". One can download for free his text on his homepage at Cornell math department, or buy it in book format. It requires that the reader has read the point set topology part of Armstrong's book.Another excellent topology book (my favorite) is Bredon's "Topology and Geometry". All the material in the above books are contained in this one. It is a quite concise book, so for an introduction to the subject Armstrong is still better. I believe that Hatcher and Bredon complement each other in a very nice way. Bredon for its precision and elegant style, Hatcher for some geometrical intuition.