A year ago this month I reviewed John Conway’s A Course in Abstract Analysis and now a year later I find myself reviewing his A Course in Point Set Topology. Perhaps, assuming the author can keep up the grueling pace, reviewing one of his books will become a New Year’s tradition for me. I certainly hope so.
Conway’s books all seem to have certain features in common which to my mind make them very attractive reading. The writing is clear, but, more than that, it is chatty and conversational; Conway has that rare and valuable ability to write reasonably informally without sacrificing precision. In addition, his books also generally seem to have, for want of a better word, personality. Conway has been a mathematician for a good while now; he has obviously formed some definite opinions about analysis, and is not shy about letting his books reflect those opinions. He also puts more of an emphasis in his books on historical background (including short biographies of many of the mathematicians whose names adorn theorems in the subject) and etymology of mathematical terms, than one typically finds in the textbook literature. Consider, for example, his thoughts about the name of the Baire Category Theorem:
Why is the word category used in the name of this theorem? Please excuse the author while he takes a little time to rant. Mathematics loves tradition; largely this is good and has my support. However, occasionally it adopts a word, makes it a definition, and promulgates it far beyond its usefulness. Such is the case here. There is a concept in topology called a set of the first category. Sets that are not of the first category are said to be of the second category. What are the definitions? I won’t tell you. If you are curious, you can look them up, but it will not be helpful. I will say, however, that using this terminology the Baire Category Theorem says that a complete metric space is of the second category. My objection stems from the fact that this “category” terminology does not convey any sense of what the concept is. There is another pair of terms that is used: meager or thin and comeager or thick. These at least convey some sense of what the terms mean. But I see no reason to learn additional terminology. The theorem as stated previously says what it says, end of story—and end of rant.
I like reading things like this in a mathematics text, and so do, I suspect, the students. Comments like this enliven a book and also educate a beginning student.
The book under review is, as the title makes clear, an introduction to point set topology, and it maintains the high quality that the author has set with his previous books. The prerequisites are modest: technically, just a good calculus background should suffice, but some prior exposure to “theoretical calculus”, as, for example, in a real analysis course (or perhaps even a really good honors calculus course), would be a definite plus.
One very distinctive feature of the book is that it is quite short (or perhaps, given the subject matter, I should say “compact”); the main text is only about 120 pages long, not counting an appendix (roughly fifteen pages long) on set theory, which starts from scratch with the definitions of “union” and “intersection” and proceeds through a discussion of Zorn’s Lemma and basic cardinal arithmetic. The brevity of the text reflects what the author refers to as a mid-career “epiphany”: “I realized I didn’t have to teach my students everything I had learned about the subject at hand. I learned mathematics in school that I never used again, and not just because those things were in areas in which I never did research. At least part of this, I suspect, was because some of my teachers hadn’t had this insight.” So, Conway wrote this book to give students “a set of tools”, discussing “material [that] is used in almost every part of mathematics.”
There are only three chapters. The first is on metric spaces and covers the basic topological concepts (continuity, convergence, compactness, connectedness, etc.) in that context. I fully agree with the author that this is the best way to introduce the subject of topology: it builds on the reader’s Euclidean intuition of distance and also provides an easy way to motivate the definition of a general topological space, which in this book is the subject of chapter 2. The topics mentioned above are revisited in this chapter in this broader context. Nets now replace sequences, which don’t work as well in topological spaces as they do in metric spaces.
There is some mild redundancy between this chapter and the first, with some results stated that are a direct generalization of the metric space result, but I certainly don’t view that as a problem; in fact, to the contrary, it seems to me to be a pedagogically sound approach to learning the material, and there’s no real waste of time, since the author, in cases of proofs that generalize almost verbatim, generally just asks the student to fill in the details. And of course there are a number of topics that are introduced in this chapter for the first time: for example, path-connectedness is defined in this chapter, and Tikhonov’s (that’s the author’s spelling; I learned it as Tychonoff) theorem is proved, using Alexander’s theorem that a space is compact if (and of course only if) any subbasic open cover has a finite subcover. (It should be noted that there is a standing convention that all topological spaces encountered in the text are assumed Hausdorff, and therefore that hypothesis is not generally repeated in statements of theorems.)
The selection of topics in chapter 3 (“Continuous Real-Valued Functions”) is based on the author’s belief that “the continuous functions on a space are more important than the underlying space”. Although many of the topics included here (regular, completely regular, and normal spaces; the Stone-Cech compactification) are typically found in other point set topology books in chapters with the word “separation” in the title, the unifying thread of the topics in this chapter is that they involve real-valued functions defined on a topological space, perhaps considered as themselves elements of a topological space. The chapter also discusses paracompactness, but stops short of proving results like the Nagata-Smirnov metrization theorem.
Slim books like this have some advantages: they don’t overwhelm and intimidate the student and they are also affordable; this one is currently selling on amazon for less than thirty dollars. (With textbook prices getting ridiculously high, I have on more than one occasion found myself considering the price of a book as a factor in the selection process.) Of course, they also have some potential disadvantages, including a certain lack of flexibility — the professor no longer gets to pick and choose from among a broad selection of things to teach from, and one’s favorite topic may well wind up missing. It’s certainly reasonable to believe that you can get a higher quality meal at a restaurant with a limited menu than at one with a mammoth buffet, but you do need to check the menu first to make sure there are items to your liking on it. So, for example, if you adamantly believe that any course in introductory topology simply must contain at least some discussion of surfaces, or an introduction to the fundamental group, or of the topological aspects of matrix groups, then you’ll probably want to look elsewhere for a course text; these topics don’t appear here. (Texts which offer a point of view different from this one include McCleary’s A First Course in Topology: Continuity and Dimension and Topology Now! by Messer and Straffin; in addition, the recent Elements of Topology by Singh, which contains enough material for two semesters, covers the topics discussed in this book and also includes some discussion of the fundamental group, covering spaces, and matrix groups, and other topics not included here.)
Another potential disadvantage of narrowly focused books like this one is that, depending on taste, one might believe that students do not end the semester with any kind of “big payoff”. The “set of tools” for analysis is provided to the student, but (particularly with respect to the topics in the latter part of the course, where the material is somewhat technical and not such a direct generalization of the analysis studied previously) he or she will have to wait for future courses to see these tools applied seriously. It’s easy to motivate the definition of a group in an abstract algebra class just by showing the students how groups pop up all over the place — geometry, number theory, analysis, and elsewhere. But it’s a little harder to get a student excited about the definition of a completely regular topological space, especially when the examples strike many of them as kind of strange and artificial.
This is all a matter of taste, of course. There are plenty of people who think this material is beautiful in its own right, and non-specialists in the area (like me) can certainly benefit from reading an expert’s opinion of what is, or is not, important. By and large, based on my own (admittedly non-expert) opinion, I thought the author did an excellent job selecting topics, with one exception — although there is a section on quotient spaces in chapter 2, there is no reference at all to either Moebius strips, the Klein bottle, or the projective plane. How can you have an introductory text on topology that doesn’t even have a picture of a Moebius strip in it? Fortunately this is hardly a deal-breaker; an instructor who is of a mind to do so can easily supplement the book on this one point. However, having already defined quotient spaces, it does seem like a missed opportunity to not give these neat examples.
Each of the three chapters is divided into sections, and each section ends with a fairly generous selection of exercises. Based on a quick survey of them, it seems that very few are of the trivial make-work variety, and most seemed to be of average or perhaps somewhat above-average difficulty. No solutions are provided in the text, which I view as another pedagogical plus.
To summarize: this is a well-written book that I enjoyed reading. Assuming that your idea of what to teach in a first-semester course in topology is in line with the author’s, this book would make an excellent text for such a course.
Mark Hunacek (firstname.lastname@example.org) teaches mathematics at Iowa State University.