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Point Set Topology

Steven A. Gaal
Dover Publications
Publication Date: 
Number of Pages: 
[Reviewed by
Marion Cohen
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I chose to review this book because I love topology, have little occasion to use it in either my teaching or my research, and wanted an excuse to be with it! Reading this book, I see that it is well-written, competent, and quite exhaustive (but including only point-set topology, as per its title, and no homotopy theory). Of course I enjoyed it! In fact, I found myself reading it the way, right or wrong, I sometimes read a novel (though more slowly), for example turning ahead or opening it to an arbitrary page. (I remember enough topology to do this, but in cases where I couldn’t, I either consulted the index or (more likely) flipped through the previous pages to find whatever definitions or theorems I needed.)

No question that this book is clear, concise, and in general worthy for its intended audience, namely advanced undergraduates and graduate students. No question that I would recommend it.

One unusual feature is that the exercises at the end of each section often contain hints as to their solutions, sometimes to the extent of virtually solving the problem. This seems unusual to me, and part of me likes this idea, or at least understands how tempting this practice could be to an author. Right now I’m looking at p. 67, Exercise 3, in the section on Products of Uniformizable Spaces (first chapter, Topological Spaces): “Let Xs for every s = 1, 2, … be the set of positive integers topologized by the discrete topology. Show by using continued fractions that the product space X = ∏ (Xs, s = 1, 2, …) is homeomorphic to the space formed by the set of irrational numbers exceeding one under the topology induced by the usual topology of the reals.” The mere mention of “continued fractions” should, I think, be enough for the creative student, but the author goes on to do much of the remainder of the work, too, leaving mostly the gory details to the student/reader. Again, if I were the author, I admit I might be tempted to do the same thing, but I’m not sure that it’s the right thing.

This was, to me, the most salient feature of the book. To explore other features, I felt that I needed to compare it to two other topology texts published around the same time. A colleague lent me his favorite, Stephen Willard’s General Topology, published in 1966, and I dug up my old copy of Kelley (also General Topology), which appeared in 1955 and from which I originally learned topology. Each text has its own way of doing things; in particular, there were more differences than similarities in the orders in which the various topics were introduced. Gaal, the book at hand, gulped down most of the basics of topological spaces in the rather long (over 50 pages) first chapter (after an introduction on set theory), including uniform structures and uniform spaces, topologies on linearly ordered sets, product topologies, and metric spaces. Willard’s second chapter (after a first chapter on set theory) is also titled “Topological Spaces”, but is shorter. Kelley’s first chapter (after an introduction on “Preliminaries”), again called “Topological Spaces”, covers mostly the basics, but ends with a section on connected sets, made more extensive by some of of chapter problems — for example, the Finite Chain Theorem for Connected Sets is included, as well as investigation of locally connected spaces.

Separation properties are treated differently among the three texts. Gaal devotes his entire Chapter II to them, and this is where he introduces connectivity (after the notion of “separated sets”; thus he treats connectedness as a separation property). Willard’s entire Chapter 5 (of 10) is called “Separation and Countability”, whereas, so it seemed to me, Kelley scatters his separation properties throughout his book; in particular T0 and T1 appear on pp. 56–7, in an exercise on accumulation points at the end of the first chapter, whereas the Hausdorff property appears on p. 67, in the section on nets, and T3 appears on p. 113.

Speaking of nets, Gaal waits ‘til his last chapter, “Theory of Convergence”, to deal with them. Both Willard and Kelley give nets a higher (or a sooner…) priority, introducing nets in their second chapters (“Convergence” and “Moore-Smith Convergence” resp.)

I have no preference for any one ordering over any other. Since Gaal is “our book”, I will briefly summarize it. Like the other two authors, he has an introduction on Set Theory. (Naïve, but including the Axiom of Choice. I should note, though, that Kelley has, in addition, an Appendix which includes a fair amount of non-naïve set theory.) Chapter I (out of five), “Topological Spaces”, begins with the obligatory definitions of open and closed sets, interior, boundary, etc., closure operators, bases and subbases. It continues with linearly ordered sets, metric spaces, filters, uniform structures, subspaces, product spaces, quotient spaces, and inverse/direct images of topologies. Chapter II, “Separation Properties”, includes them all (in particular, as I’ve said, connectedness) as well as separable spaces and countability axioms. Chapter III is “Compactness and Uniformization”; it includes separation properties of compact spaces, Tychonoff’s Theorem, locally compact spaces (and their products), the equivalence of paracompactness and full-normality, Urysohn’s theorem (that a  space is pseudometrizable if and only it it has a base which is the union of countably many locally finite systems), and metrizability conditions.

Continuity is introduced in Chapter IV, in particular how it relates to the axioms of separation, compactness, connectedness, and product spaces, also the (for some surprising) fact that the closed interval [0,1] is a continuous image of the Cantor set. Uniform continuity and convergence (relative to given uniformities) are penultimate in this chapter, which ends with the Weierstrass Theorem on the approximation of continuous functions by polynomials, as well as its generalization by M. Stone. The Theory of Convergence is saved for the last Chapter V; one reads about nets/filters, universal nets/ultrafilters, and how they relate to products, compactness, and Cauchy and complete spaces. Also appearing are the Baire Category Theorem, the Principle of Uniform Boundedness and the Condensation of Singularities. Completions and compactifications end the book.

I have two main criticisms. First, and more important, as far as I could find, there was no motivation for the defining, in the first place, of a topological space in terms of those familiar properties of open subsets. True, one can soon infer the reason as soon as one is introduced to the example of the reals with the usual topology, but … well, when I took my first course in topology, that wasn’t sufficient for me; I still wondered what the properties of open sets and all else that they led to had to do with “rubber sheet geometry” and “Why is a donut like a coffee cup?” So I feel that it’s important to motivate before plunging into the definition of a topological space. (And here I just have to make quick mention of the great motivation in Willard’s book, beginning on p. 17 in the section introducing metric spaces: “…can we eliminate the dependence, in the previous definition [of continuity of functions between metric spaces], on the presence of distance functions? The answer is affirmative and depends on the development of the notion of an open set in a metric space.” Actually, I would also move on from continuous maps to homeomorphisms, so that first-time topology students could see that “a donut is like a coffee cup” because there is a homeomorphism from one to the other — likewise one “rubber sheet” being moldable into another.)

The second criticism is probably more of an annoyance. Many deep results have, of course, titles (in any subject), the last word being “Theorem” (or sometimes Lemma) and the other words being the names of their author(s). In this book, authors are indeed given credit for their results (Baire Category Theorem, Urysohn’s Theorem…), but only within the body of the text and not where the theorem is actually set off and stated. So in a section with several theorems and lemmas (as are most sections), it‘s difficult for a student seeing a given result for the first time to know just which theorem/lemma is, say, “Urysohn’s Theorem”, and which are auxiliary results (especially since this particular book sometimes places the main results before the auxiliary lemmas/theorems). That can be confusing. Along this same line, definitions are too often not set off.

All three books merited being read the way I read Gaal, namely as described above, like a novel (the way I read novels…). But I have to confess that, probably because of its main difference from the other two — namely, smaller print and smaller vertical spaces between lines — but perhaps also because the book contains precisely one diagram — I found Gaal less pleasurable to read and probably would not choose it as a text, were I to teach a course in topology. These considerations are very likely minor, so ultimately, making a choice among the three is, to me, like making a choice in an Italian restaurant among spumoni, tortoni, and rum cake or, back to novels, in a Russian literature course, among Anna Karenina, War and Peace, and Crime and Punishment.

Marion Cohen teaches math at Arcadia University in Glenside, PA. She’s particularly excited about a course that she developed and taught last year: “Truth and Beauty: Mathematics in Literature.” She is the author of Crossing the Equal Sign (Plain View Press, TX), a collection of poetry about the experience of mathematics. Another book, Chronic Progressive, is forthcoming from the same publisher. This book is not specifically about math, but it contains many math-inspired poems.

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