Understanding Topology

Topology is one of those subjects that can be taught to undergraduates in a number of different ways. I learned it, as an undergraduate, from the first part of Simmons’s book Introduction to Topology and Modern Analysis; the course followed the usual point-set route of metric spaces, topological spaces, compactness and connectedness. I thought it was a great course at the time, because it helped put real analysis into what I thought was a better context, but only afterwards came to realize that there was much more to the subject, even at the undergraduate level. Simmons’s book is weak on the geometric aspects of topology, and not only are things like surfaces not mentioned, neither are quotient spaces; in fact, there is nary a Möbius strip or Klein bottle to be found in the book. Also, Simmons (like most other undergraduate topology texts at the time) doesn’t address basic algebraic topology at all.

Some of the more current undergraduate topology texts take different approaches to the subject. Adams and Franzosa, in Introduction to Topology: Pure and Applied, pay a lot of attention to applications of topology. McCleary’s A First Course in Topology: Continuity and Dimension contains a lot of the traditional point-set stuff but is geared towards the overall goal of proving invariance of dimension, so there is a lot of geometric topology done as well, and also some discussion of the fundamental group and homology. Messer and Straffin, in Topology Now!, emphasize things like surfaces, knots and manifolds, with metric and topological spaces, per se, not appearing until the end of the book, and even then, only quite briefly. Messer and Straffin also discuss the fundamental group, as do, in a more traditional point-set framework, two Dover paperbacks by Gemignani (Elementary Topology) and Croom (Principles of Topology).

It seems clear, therefore, that any instructor of an undergraduate topology course must make some difficult choices as to topic coverage and textbook selection. The book now under review seeks to make these choices easier, by providing a look at the subject that touches on quite a lot of topics.

The back cover of the book says that it “requires only a knowledge of calculus and a general familiarity with set theory and logic.” I’m not sure I agree. Several proofs fall back on the kind of familiarity with the real numbers (e.g., completeness) that one learns in a course in analysis, and some prior background in abstract and linear algebra is probably necessary to really get much out of Part III of the text, described in more detail below. There is a brief appendix on groups and linear algebra, but as is inevitably the case with appendices of this type, it does not replace the kind of understanding and insight that comes from a prior course in the subject.

It should also be included that the “familiarity with set theory” referred to above does not mean a familiarity with cardinal arithmetic. The author tends to shy away from this topic; he includes a 30-page appendix on set theory, functions and equivalence relations but does not cover countable or uncountable sets, and avoids their use in the text. So, for example, the co-countable topology is not mentioned as an example here. Likewise, Zorn’s lemma and the axiom of choice are neither discussed nor used.

The main body of the text is divided into three parts, each one consisting of several chapters. The first part is on Euclidean Topology. It begins with an introductory chapter that gives a broad summary of what topology is about, including the now-obligatory discussion of a coffee cup transforming into a doughnut. The next chapter is largely on metric topology in Euclidean space, with a section at the end describing how these ideas generalize to arbitrary metric spaces. The final chapter in this part is on vector fields in the plane. A lot of the material in this chapter would not be out of place in a differential equations textbook, but Ault’s goal is not to study ODEs but to use topology to prove global results about phase plane portraits. A highlight of this chapter is a proof of the Poincaré-Bendixson theorem.

Part II is entitled “Abstract Topology with Applications” and opens with a chapter on general topological spaces, reviewing and generalizing many of the ideas discussed previously. (Kudos to the author for including a section here on Furstenberg’s nice topological proof of the infinitude of primes; for fun, he also throws in Euclid’s proof.) This is followed by a chapter on surfaces, which among other things introduces the idea of the Euler characteristic. The applications referred to in the title of this part concern graphs and knots, and are discussed in the next chapter.

Part III is an introduction to algebraic topology, covering first the fundamental group and then (more surprisingly, since this topic doesn’t get covered much at the undergraduate level) homology — first rational homology, then integral homology. This is, I think, the hardest part of the book, and likely the least successful. The discussions here are similar to, and not dramatically easier than, those found in the first few chapters of Hatcher’s algebraic topology book, which is very much a graduate-level text.

Fundamental groups do, as previously noted, appear in several undergraduate-level texts, and I think that’s a good thing, but it seems to me that a little bit of this material, with emphasis on examples and intuition rather than proof, goes a long way in a first undergraduate course. (I taught a first course in topology a few semesters back and considered myself lucky to get to a motivated but precise definition of homotopic curves, and a couple of proof-by-picture calculations of fundamental groups.) The author here takes this topic somewhat further than one might expect, not only discussing universal covering spaces and applications to knot theory, but also throwing in an optional section on higher homotopy groups. This is likely much more than can be comfortably covered in a first course.

Then there are homology groups. I have never believed that they are really an appropriate subject for most undergraduates (unless you want to dispense with proofs altogether and just discuss them in very vague, hand-waving terms), and I’m afraid that this chapter has not changed my opinion. The author tries very hard to make the ideas intuitive, but this chapter (particularly the second section on integral homology) involves a lot of fairly technical material (including modules) and overall it represents, I think, a sudden and rather jarring increase in level of difficulty from the rest of the text.

Of course, the final chapters of a text are ones that are easy to skip, and overall, this is an

attractive candidate as a text for an undergraduate course. The variety of topics covered is certainly one reason for this. Another is the author’s obvious concern with good pedagogy. Examples and illustrations abound. So do exercises — some are completely trivial, but some are more interesting. (Solutions to a number of them appear in the book. This would actually not be my choice, since it makes it harder for an instructor to assign homework, but it does have the advantage of increasing the value of the book for self-study.)

The writing style is clear, conversational and generally accessible, and conveys a sense of enthusiasm for the material. The level of rigor throughout the text varies, and is generally appropriate to the topic being covered — some things are proved rigorously, others are not, and some proofs (such as that of the Jordan Curve theorem) are simply omitted altogether. When the author doesn’t do something rigorously, he makes this fact clear, so that at no point is the student reader ever misled as to what is a precise argument and what is not.

Covering a lot of different topics, however, can be both a blessing and a curse. Because of the variety of topics covered, not all topics are covered to the same degree that they might be in some other books. This is true, for example, in the case of metric space and topological space theory, and instructors who believe that a first course in topology should give students a thorough grounding in these topics might find the discussion here lacking in some respects.

For example, although Ault defines sequential compactness (for topological spaces, even though sequences don’t really work well in arbitrary topological spaces) he never, as far as I can tell, mentions, let alone proves, that sequential compactness is equivalent to compactness for metric spaces. A few pages before discussing sequential compactness, he mentions (in a brief footnote, without proof) that limit point compactness is, in the context of metric spaces, equivalent to compactness, but he does not tie that in with sequential compactness. He also does not prove the other standard equivalent condition (complete and totally bounded) for compactness in metric spaces.

Also, although the notions of manifold and simple connectivity are mentioned, he does not discuss the Poincaré conjecture and its interesting history, which seems like a missed opportunity.

Other omissions that might bother some instructors include the lack of any discussion of

• separation axioms
• a characterization of connected subsets of the real line
• uniform continuity and its relationship to compactness
• the Baire category theorem
• the one-point compactification
• simple applications of the intermediate value theorem (such as, for example, the simple result that a continuous function from a closed bounded interval of the real numbers to itself has a fixed point)
• the contraction mapping principle and its applications
• matrix groups as topological spaces
• metrization theorems

Obviously it would be a rare course in which all of these topics were covered, but an instructor might want to discuss some of them.

In the final analysis, of course, the question of whether these omissions are justified by the inclusion of the other topics mentioned above is a matter of individual preference. I think, however, that there are several things that the author could have done to find room for at least some of these omitted topics. For one thing, he could have skipped the chapter on homology. Also, if chapter 2, instead of being mostly about metric concepts in Euclidean spaces, had started out with the definition of an arbitrary metric space, and made frequent forays into Euclidean space as illustrative examples, the chapter would have been more streamlined and might have allowed for a few extra sections. I realize that there may be some pedagogical benefit in starting out with something concrete and then circling back to repeat these ideas, but I also think that an undergraduate who is sufficiently far along in his studies to be taking a course in topology should be able to handle the general notion of a metric space right off the bat.

I have a few other, much less substantive, quibbles. First, since there is more material here than can be covered in a semester, a chapter dependency table, or discussion in the preface about which chapters depend on which and to what extent, would be helpful to an instructor planning a course based on this text. Second, although there is a pretty good index, it could be punched up in several regards. A reader looking at the index, for example, will not be able to conveniently find the definition of the closure of a subset of a topological space, or a statement of Tychonoff’s theorem on product spaces (which the author proves for finite products and states for general products), or the proof of the Poincaré-Bendixson theorem.

To summarize and conclude: the back cover of this book describes it as the “perfect textbook” for an introductory topology course. It’s not, of course; I don’t think that a perfect textbook — for topology, or anything else — exists. It is, however, a book with an interesting take on the subject and which, as described earlier, has much to recommend it. If you are teaching an undergraduate course, want an interesting selection of topics to choose from, and are willing to dispense with some depth in the discussion of metric and topological spaces, then this book is well worth a serious look.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.