A First Course in Topology: Continuity and Dimension

How shall we introduce undergraduates to topology? When I was an undergraduate, I took a course that resolutely focused on general topology, beginning with the usual definitions and ending up in a morass of separation axioms and weird examples. I found the first half of the course rather easy, since notions such as connectedness and compactness had already been introduced in analysis courses and it was just a matter of straightforward generalization. I found the second half boring, because there were simply no interesting examples. My mathematical background was insufficient to allow me to understand any of the real applications of what I was learning, and it looked to me like an endless amount of logic-splitting.

This experience seems to be fairly common. As a result, several authors have tried to come up with a new way of introducing undergraduates to topology. Two recent examples are Topology Now!, by Robert Messer and Philip Straffin, and Topology, by Sheldon Davis. These take different approaches: Messer and Straffin focus on algebraic topology (broadly speaking), while Davis sticks closer to general topology, but emphasizes different aspects of it. The book under review is a third attempt, one that I find quite successful.

McCleary has chosen to order his course around a specific problem, the problem of invariance of dimension. Once Cantor had proved that spaces of various dimensions were set-theoretically indistinguishable, it was natural to ask whether there was any substance to the notion of dimension. In what sense was two-dimensional space different from three-dimensional space? Some forty years passed between Cantor's set-theoretic discovery and Brouwer's proof that in fact Rⁿ and R^m can be homeomorphic only if m = n.

A First Course in Topology opens with Cantor's proof, and concludes with a homological proof of Brouwer's result. Along the way, McCleary lays down the foundations of general topology, develops the theory of the fundamental group, glances at covering spaces, proves the Jordan curve theorem, and finally introduces simplicial homology to establish invariance of dimension. Everything is done efficiently and elegantly.

McCleary's book is packed with fascinating material, explained clearly but very succinctly. He is not afraid of hard theorems. To show how delicate the problem of dimension is, he introduces Peano's space-filling curve and proves that it is both continuous and surjective. To introduce the idea of using combinatorial approximations to topological problems, he proves the Jordan Curve Theorem — not a proof often seen in textbooks. And the idea works, too. Not being a very good combinatorialist, I have always been somewhat allergic to the simplicial approach to homology. McCleary kept me reading, and actually helped me understand how and why this approach can be useful.

In order to get to his target result, McCleary sometimes needs to make hard decisions. Unless some topics are omitted, there won't be space to get to the big theorem, but nothing that needs to be used in that theorem can be omitted. Among the hardest of these choices must have been the decision to avoid the issue of orientation entirely, and (in order to be able to do that) to do all his homology theory in characteristic two. I think this works well, and enough signals are given to the reader to alert him to the fact that there is more to the story. A final section points to various choices of where to go to learn more.

The pre-requisites for the book are fairly heavy for an undergraduate text. Some abstract algebra is required, especially for the fundamental group and for linear algebra over F₂. McCleary motivates the notion of a topological space using the ε–δ definition of continuity, so that students will need to feel fairly comfortable with that. At my college, this would mean that the book could be used only with students who had taken both the abstract algebra and the real analysis course. Given the background, however, this First Course would make a wonderful capstone to an undergraduate major, because it builds so effectively on the ideas from different areas of mathematics to get to something substantial. It would also be useful for first-year graduate students, provided they do not want to (and are not required to) jump straight into algebraic topology.

Reviewers are supposed to pick nits, but I have very few. I don't like the way McCleary puts names of mathematicians in small caps when they first appear; for me, it makes them jump out from the text a little too much. In the chapter on compactness, my Bourbakist training reared its head to protest that "compact" should mean that a space is both Hausdorf and has the finite subcovering property. Since McCleary defines compactness by requiring only the subcovering property, the assumption that various spaces are Hausdorff has to appear in a lot of theorems. Both complaints are, of course, minor. Generally speaking, McCleary has done a really careful job.

Like most in the AMS Student Mathematical Library series, this book is physically small, though at 210 packed pages it is really larger than it seems. Still, I think it is adequately described as a little book, since it fits in the hand much more comfortably than most textbooks. It repeatedly brought to my mind one of my favorite mathematics books, Spivak's Calculus on Manifolds: physically, in its approach to mathematics, and in its ability to hold the readers interest.

This is a beautiful little book that may well become a classic. I learned some things as I was reading it; students who work through it will learn a great deal, and emerge from the process much better mathematicians than they were before they began. It deserves many such readers.

Fernando Q. Gouvêa teaches at Colby College in Waterville, ME.