Michael Gemignani has led an interesting life — he is an Episcopal priest, lawyer and mathematician, and has written books in all three areas. Back in the late 1960s and 1970s he wrote quite a few books in mathematics (including ones on topology, real analysis, axiomatic geometry, probability and calculus); in the 1980s he wrote several law-related books, including at least one on computer law; more recently he has written on spiritual topics, both fiction (What Really Happened to Harry?) and nonfiction (Paths to Contemplation). He has even written a book (unseen by me), Farnsworth, about a near-bankrupt university facing an offer from Satan.
The book now under review is a Dover reprint of the second (1972) edition of his topology book; Satan does not appear, but lots of topics do that have bedeviled many students over the years. I recall flipping through the first edition of this book (then published by Addison-Wesley) as supplementary reading in an undergraduate topology course that I was taking around 1970; I thought at the time that it was a good book, and still do, although, as explained below, I do have some concerns about a couple of points.
The topics covered here are, for the most part, ones that are fairly standard in introductory topology courses: after a preliminary chapter on sets and cardinality (and also a brief discussion of groups, in anticipation of the final chapter on the fundamental group), Gemignani offers a chapter on metric, and then two on topological, spaces, each one covering the basics. (I strongly believe that this is the proper order in which to do things; metric spaces provide a natural generalization of the student’s work in calculus and analysis, and topological spaces are a natural generalization of metric spaces; this provides a well-motivated path to what might otherwise be a strange concept.) This is then followed by a chapter on the separation axioms.
The next chapter (“Convergence”) is a particularly nice, well-motivated account of general notions of convergence (filters and nets), beginning with a few pages explaining why sequential convergence is not terribly useful in general topological spaces. The author could also have pointed out a much simpler example: in an indiscrete topological space, every sequence converges to every point in the space, so for that reason alone the notion of sequential convergence seems to be of limited utility.
One standard application of these ideas is a proof of Tychonoff’s theorem on products of compact spaces, and the author gives this as part of a discussion, comprising the next three chapters of the text, on compactness and connectedness. The version of Tychonoff’s theorem presented in the main body of the text is for countable products only, and the author also gives an alternative proof of this theorem that does not involve filters and nets. In an Appendix, he discusses arbitrary products and proves the result in that generality.
The final two chapters of the book address, first, metrizability and complete metric spaces, and then homotopy theory. The first of these chapters, among other things, proves the Urysohn metrization theorem and discusses compactness in metric spaces. In the second of these chapters, the author defines the fundamental group, provides some simple examples, and proves some basic theorems (e.g., the fundamental group of a product space, the mapping on fundamental groups induced by a continuous function between spaces). This chapter stops short of being a full-blown discussion of this topic; covering spaces are not mentioned, for example, and neither is the Seifert van Kampen theorem. Gemignani’s decision not to discuss these topics seems entirely reasonable for a first course in basic topology that is not intended to go very far into algebraic topology. There are some books, like Munkres’ Topology, that go deeper into the fundamental group, as do books like Singh’s Elements of Topology and Introduction to Topology by Gamelin and Greene; these texts all contain enough material for a two semester course, however.
There are, however, some other omitted topics that I would prefer to have seen included: Although quotient spaces are defined (the author calls them identification spaces, which seems a bit old-fashioned to me), familiar spaces like the Moebius strip and Klein bottle are not mentioned at all. It seems hard to imagine an introductory text on topology that does not have a picture of a Moebius strip in it, but this is by no means an isolated phenomenon; the book I learned this material from, Simmons’ Introduction to Topology and Modern Analysis omits it, as does the much more recent A Course in Point-Set Topology by Conway.
In addition, and this may simply reflect my own personal prejudices, I like the idea of discussing, at least briefly, the topology of some of the more basic matrix groups, but these are also not mentioned; perhaps, when the book was originally written, they were not viewed as being as important as they are now.
I have a few other quibbles, the first of which is not the fault of the author: perhaps the fault lies with my increasingly elderly eyes, but I thought the typeface in the book was rather small, particularly with regard to page numbers; on several occasions I had trouble determining what number a particular page was.
On a more substantive level, there are two issues that I think merit discussion. First, the author is sometimes a bit loose with terminology. The term “path in X”, where X is a topological space, is first defined (page 111) as a continuous function from [0 ,1] into X; later, however, on page 189, the exact same phrase is defined anew to mean that X is the image of such a function. It is this latter definition that the author uses when he later gives, without proof, this statement of the Hahn-Mazurkiewicz theorem: “A metric space is compact, connected, and locally connected if and only if it is a path.”
Another example along these lines occurs in section 3.5, where the term “derived set” is given both its usual meaning in point-set topology (the set of limit points of a set) and is also used in the section heading to refer to sets “which are topologically related to or ‘derived’” from a given set A.
While these examples of linguistic imprecision are probably easily fixed in lecture, there is a second issue that I think is more problematic. The important concepts of compactness and connectedness are not introduced until quite late in the text, following the chapters on separation axioms and convergence; it seems clear to me, though, that it is more important for a beginning student in topology to know what a connected or compact space is, then it is for him or her to know the difference between a T3 and T4 space. In fact, I think that the concepts of connectedness and compactness are so fundamental to a good understanding of elementary topology that they should have been introduced initially in the metric space chapter, thus giving the reader adequate time in which to absorb and assimilate these ideas. Likewise, the important concept of a complete metric space is not discussed until very near the end of the book; this is another topic that could (and in my opinion should) have been brought up earlier, when metric spaces were first discussed.
On the plus side, there are certainly aspects of this book that I liked. The author’s writing style is clear and generally well motivated, and the exercise collection is a good one, containing quite a lot of exercises, many of which are just challenging enough to be interesting but not unreasonable. Working through them would be a valuable activity for students. (Solutions are not provided in the text, and I see no indication of a solutions manual being available.) And of course the very inexpensive price of this book (about seven dollars on amazon, as I write this) is a huge inducement to use this as a text instead of a book that costs twenty times that price.
Whether these positive aspects of the book outweigh some of the concerns expressed previously is, of course, a matter of personal taste; in any event, I think this text definitely merits serious consideration as a possible text for an undergraduate-level text in topology. And, at this price, it is certainly a book that a student or instructor might just want to buy as a reference.
Mark Hunacek (firstname.lastname@example.org) teaches mathematics at Iowa State University.