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Principles of Topology

Fred H. Croom
Dover Publications
Publication Date: 
Number of Pages: 
[Reviewed by
Mark Hunacek
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This is a very nice introduction to (mostly) point-set topology, ideally suited for use as a text in courses where the students have diverse backgrounds and a very sophisticated treatment is unwarranted. The exposition here is careful, conversational, clear, and well-motivated. The only real background required is a good working knowledge of calculus and some practice in writing and reading proofs. Examples abound, as do exercises of reasonable difficulty. There are also nice (albeit brief) historical discussions at the end of every chapter except the first, which is largely introductory. And, of course, as a Dover paperback, the price is right (less than fifteen dollars, as of this writing).

The order of presentation of the material also has some pedagogical advantages. I have long thought, for example, that the best way for students to be introduced to topology is by doing metric spaces first. It is, after all, a reasonably small jump from working with absolute values on the real line to the more abstract idea of a distance function, so metric spaces are easy to motivate. They, in turn, then help motivate the more abstract notion of a topological space, because by the time topological spaces are ready to be defined, the student has already seen open sets in metric spaces and is familiar with their properties.

That is the approach taken here. The first chapter is introductory in nature. It covers some very basic material on sets, functions and equivalence relations, but, more importantly, starts with a well written heuristic introduction to the nature and history of topology. Students even get to see a Mobius strip right away. The exposition in these opening pages is of course non-rigorous, but later in the book surfaces and manifolds are discussed again as quotient spaces. (This is, to my mind, a very positive feature: some books on point-set topology (see the discussion below) downplay this aspect of topology, subjugating it to the analytic aspects of the subject.)

The next chapter discusses the basic topology of the real numbers and the plane, and also discusses countable and uncountable sets. Cardinality is not gone into in great depth, but it is at least established that the set of rational numbers is countable and the set of all real numbers is not. The topological discussions include open and closed sets, limit points, and so on. The Heine-Borel theorem for closed and bounded intervals, for example, is proved, but without using the word “compact”: the statement in this chapter is that any open cover of a closed and bounded interval contains a finite subcover. Later on, in the chapter on compact topological spaces, the “full” version of the theorem (that a subset of Euclidean space is compact if and only if it is closed and bounded) is stated and proved. Students who have had a previous course in real analysis can probably skim this chapter, or skip over it altogether.

Chapter 3 introduces metric spaces as a natural generalization of the notion of distance on the line and in the plane. The basic definitions and properties of metric spaces (open and closed sets, sequential limits, continuity, etc.) are discussed in detail, with lots of examples. Complete metric spaces are introduced, and both the Baire Category Theorem and the Banach Contraction Mapping Principle are proved. The author states that these results “have wide applicability” and mentions a few applications, but without any attempt at proof. Given the intended audience of this book, this is probably not unreasonable.

Chapter 4 is devoted to topological spaces, and discusses the standard concepts relating to them: closed sets, interior, closure and boundary; continuous functions and homeomorphisms; bases and subbases; and subspaces. It is mentioned that sequences don’t behave as well in arbitrary topological spaces as they do in metric spaces, but nets are not defined.

Connected and compact topological spaces are the subjects of chapters 5 and 6, respectively. In addition to establishing the usual properties of these spaces, the chapter on connected spaces also discusses path connectedness and local path connectedness; the chapter on compactness introduces the one-point compactification. One nice feature here is that the topologist’s sine curve is shown in some detail to be connected, but not path-connected; this is often left as an exercise in other texts. Another nice feature is the inclusion of an entire section on the Cantor set, giving both its definition and proving some of its basic properties. This is also a topic that is often relegated to the exercises in other books, if it is mentioned at all.

Chapter 7 discusses two ways to create new topological spaces from old ones: products and quotients. With regard to product spaces, Tychonoff’s theorem on the product of compact spaces is proved for arbitrary, not just finite, products. This theorem, in full generality, is equivalent to Zorn’s Lemma, which is not discussed in the text; the author finesses this issue by assuming the Alexander Subbasis Theorem, the proof of which, he says, “involves set theoretic considerations which would take us rather far afield.” He does provide an outline of a proof as an exercise, where it is pointed out that the solution “will involve some form of the Axiom of Choice” — a statement that may confuse students, since the Axiom of Choice has not previously been mentioned in the text. (The author makes it easy for readers to avoid all these issues altogether by first proving the result in the finite case and leaving the infinite product case for the next section, where it can easily be omitted.)

Chapter 8 deals with the related ideas of separation properties and metrization theorems. Normal, regular and completely regular spaces are discussed, and the standard “big” theorems in the area (Urysohn’s Lemma, Tietze Extension, Urysohn Metrization theorem) are proved. The more difficult Nagata-Smirnov metrization theorem is stated but not proved.

The final chapter of the book is a brief introduction to algebraic topology via the fundamental group of a topological space. (For readers not already familiar with the notion of a “group”, a brief Appendix discussing this topic is provided.) As an application of the fundamental group, the Brouwer fixed point theorem is proved, and the chapter then ends with a section (largely unnecessary, I think, and easily omitted) on categories and functors.

This final chapter, I thought, was quite well done: the author goes far enough into the subject to give some interesting examples and prove some nice theorems, but not far enough to be confusing to beginning students. Covering maps, for example, are discussed only to the extent necessary to prove that the fundamental group of the circle \(\mathrm{S}^1\) is isomorphic to the group of integers. (Croom has also written a book, Basic Concepts of Algebraic Topology, that purports to make that subject accessible to undergraduates; at least, it appears in the Springer Undergraduate Texts in Mathematics series. I have not seen this book, however.) One minor quibble about notation: the author uses the upper-case \(\Pi_1\) to refer to the fundamental group, but just about every other book that I know of uses the lower-case \(\pi_1\).

There are a few other stylistic tics in this book that I could quibble over as well. All proofs are written, in their entirety, in italics, which I found mildly annoying, particularly when the proofs went on for a full page or more. Also, the author uses the term “Hilbert space” to mean not the general concept of a Banach space with norm induced by an inner product, but the specific Hilbert space of square-summable sequences of real numbers. I believe that I may have seen this convention used in other books, but I think students would be better advised to think of this particular space as one example of a general concept.

These are, of course, rather minor issues, and do not seriously detract from the value of this book as a text. In fact, over the years I have had the pleasure of reviewing a number of other topology texts for this column: the classic by Munkres, other Dover reprints by Gemignani and Gamelin and Greene, and more recent books by Conway, Singh, and Manetti. These are all very good books, but looking at them solely as candidates as a text for a junior-level undergraduate course, I would put the book now under review at the top of the list. Munkres is, as I mentioned, a classic, but I think it is too difficult for use as a text, at least at my university. So is Gamelin and Greene, half of which is devoted to algebraic topology; only about a hundred pages is spent of metric and topological spaces. Conway is written in a style that is very much to my personal taste, but is very succinct and pared down (just 118 pages of text, not counting appendices) and omits material on things like manifolds, Klein bottles and Mobius strips. Manetti and Singh both have the advantage of talking about matrix groups, but seem to be not quite as reader-friendly (not quite as “handhold-y”, if that’s a word) as the text under review; Manetti and Munkres also do not start with metric spaces but instead plunge directly into topological spaces, an approach that, as I indicated above, I do not favor.

In addition to the books mentioned above, I should also cite Simmons’ Introduction to Topology and Modern Analysis, a book for which I have considerable sentimental attachment: I learned topology from this text as an undergraduate, about 45 years ago, and loved it. It is still an excellent book, written with great style and clarity, but I now think it has flaws that I didn’t notice as an undergraduate (no discussion of quotient spaces and things like Mobius strips, for example). Croom’s book is about as clearly written as Simmons, but remedies some of these flaws, so I would probably rank this text even higher than Simmons.

The preface to Croom’s book states that it is also suitable as a text for a beginning graduate course, but there I must respectfully disagree: this book is simply too easy for use as a text at that level. Better choices might be the books by Munkres and Singh mentioned earlier, or An Illustrated Introduction to Topology and Homotopy by Kalajdzievski.

This text is a Dover reprint of a book first published by Saunders in 1989. Although more than 25 years old, it still holds up very well. As a possible text for an undergraduate introductory topology course at the easy end of the spectrum, this book merits a very serious look. At fifteen dollars, it’s a steal.

Mark Hunacek ( teaches mathematics at Iowa State University.

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