A Taste of Topology

If I may, I should like to start this review of Volker Runde’s A Taste of Topology with an allusion to my earlier review of A Topological Apéritif, by Stephen Huggett and David Jordan, appearing in Read This!, on 22 April, 2004. In that review I stated that “[the Apéritif] would … play an important role in an introductory course as a supplementary text to one of the established players,” and I went on to single out Munkres’ standard text as a good choice for “the main course.” At the cost of taking word-play to altogether revolting heights, let me note now that if a somewhat more exotic and possibly a little lighter meal is desired the Apéritif might be followed by A Taste of Topology to comprise a very nice introductory course (or two) in topology: both books are easy to digest and leave the reader satisfied. (Having gotten this last pun out of my system, I will now get on with a more proper and specific consideration of Runde’s commendable book.)

A Taste of Topologygrew out of the author’s lecture notes for his 2004 senior-level course at the University of Alberta, in which he evidently tried to solve a pretty sticky pedagogical problem. “There is a very real danger,” says Runde in the preface to his book, “that students come out of a topology course believing that freely juggling with definitions and contrived examples is what mathematics — or at least topology — is all about.” Of course, as Runde goes on to observe, the logistical problem is that introductory topology offers particular difficulties because beginning students invariably lack the background and sophistication needed to fathom topology’s more “natural” (or at least canonical) examples, which often come from subjects not included in the usual undergraduate sequence.

Runde does something very interesting and, I think, very useful, in the book under review: he explicitly plays down what accordingly might to the beginner appear synthetic and artificial, and plays up material which resonates with other, already familiar, mathematical notions from, for example, analysis; and he goes well beyond what is ordinarily found in a first topology course. So it is, for instance, that in the chapter titled, “Systems of Continuous Functions,” a marvelous discussion of Urysohn’s Lemma is immediately followed by a treatment of Stone-Cech compactification and the Stone-Weierstrass Theorem. Runde gives “Silvio Machado’s short and elegant” proof of the complex form of the latter theorem.

Other perhaps unusual topics in the book include the Arzelà-Ascoli Theorem, which Runde characterizes as “the right substitute for the Heine-Borel theorem in spaces of continuous functions,” having shown early on that Heine-Borel breaks down outside Euclidean n-space. (He also appends a nice discussion of the general failure of Heine-Borel in infinite-dimensional spaces.) The highlight of this presentation is the proof of the equivalence of having a normed space be finite dimensional, of having its closed unit ball be compact, and of having each closed and bounded subset be compact. Appearing at the end of the book in the form of an appendix this elegant characterization is a nice encore to what came before.

I should also like to draw attention to my favorite chapter of the book, namely, Chapter 5, “Basic Algebraic Topology.” It is a very thorough treatment everything (appropriate at this level) from homotopy to covering spaces, with Brouwer’s fixed point theorem featured. The chapter (and the book itself) ends with an allusion to a space with a non-abelian fundamental group. Furthermore, this section also contains a truly amusing misprint: Poincaré’s Analysis Situs is given the publication date of 1985 (which may of course be evidence of the great man’s immortality).

A Taste of Topology is also rich in exercises of varying degrees of difficulty and contains excellent historical material, primarily contained in the sections labeled “Remarks” at the ends of all chapters. One particularly noteworthy historical aside occurs on p. 59 where Runde takes on Bourbaki. To wit, earlier, on p. 47, Bourbaki’s Mittag-Leffler theorem (as a prelude to Baire’s theorem that the intersection of a sequence of dense open sets in a complete metric space is again dense) is given an altogether accessible form: “Suppose that {X(n), d(n)} [n = 1, 2, 3, …] is a sequence of complete metric spaces, and let [each] f(n): X(n)  → X(n-1) … be continuous with dense range. Then [the intersection of all the iterates] f(1)f(2)…f(n)(X(n)) [for all n] is dense in X(0).” Twelve pages later Runde notes that ‘[f]or good reason, our theorem … is somewhat less general than the result from Bourbaki. As Jean Esterle remarks … ‘Incidentally, the reader interested in the French way of writing a result as clear as [this one] in a form almost inaccessible to the [human] mind is referred to the [original] statement by Bourbaki …’” Manifestly Runde possesses the gifts of understatement and a light touch which contributes substantially to the book’s readability.

Coming back to my erstwhile review of A Topological Apéritif, I also mentioned there that I should certainly choose that book “at the very least as a supplementary text” the next time I am asked to teach topology. I will now amplify this by noting that I intend to use Runde’s A Taste of Topology as the main textbook for that course. I highly recommend the book.

Michael Berg is Professor of Mathematics at Loyola Marymount University in California.