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Topology and the Language of Mathematics

Bobo Strategy
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I’ve been writing book reviews for the MAA for a little over a year now — in between acts of the nightmare the rest of my life has been. It’s been one of the few pleasant experiences I’ve had in that time. By and large, all the books I’ve reviewed have enriched my growing knowledge of mathematics and my own nascent opinions of its myriad branches and how they should be presented to beginners. Accordingly, to varying degrees, all the reviews I’ve done to this point have been positive. I’ve yet to deliver a negative review here and was hoping to continue a perfect record of positive reviews at this space. Sadly, I have to report my perfect record ends here.

This is a book on point set topology for undergraduates. Point set topology is a subject that, to me, is to mathematics what chipped beef is to culinary circles: an old standby that in earlier times, was much loved and cherished, but that due to a combination of changing values, a perceived “blandness” compared with other, flashier and more versatile dishes, as well as poor execution of late, has fallen out of favor and is spoken of only in the most derogatory of terms.

This to me is a very sad state of affairs. For me and for many mathematicians, point-set topology was among my first exposures to truly sophisticated mathematics. It is a subject whose very simplicity gives it a power and beauty that still fascinates me — it is amazing how many deep results one can prove armed simply with logic, basic set theory, and a cursory understanding of analysis on the real line. And, of course, a deep understanding of analysis simply isn’t possible without a fairly good grasp of compactness, connectedness, open and closed sets and related notions. You can study analysis without it, of course, and in the current climate many students do — but the resulting construct is ridiculously complicated; one bemoans the lack of the language of point set topology, which would simplify it and provide an organizing framework for the many different concepts.

I first learned the bare essentials of this beautiful subject as an undergraduate from my mentor, Nick Metas. I later sat in on the graduate course on the subject taught by Gerald Itzkowitz from his deep and beautiful notes and I finally mastered the subject in the course of John Terilla. I consider myself doubly blessed in this regard, since I learned the analytic aspects of the subject, such as topological groups and generalized convergence from Itzkowitz and the more geometric aspects such as quotient spaces and homotopy, from Terilla. While Terilla’s course probably prepared me for a graduate algebraic topology course better, I feel my understanding of analysis was greatly strengthened by Itzkowitz’s approach and it was fundamental in forming my attitudes towards to subject.

To me, it is analysts who really understand the subject and see its true significance.  A Riemann sum or a Lebesgue integral limit becomes a whole lot easier to picture when viewed as a convergent net over an ordered subset of the real line. The idea of convergence itself — so fundamental in analysis — becomes much easier to picture. Stephen Willard, in his classic text, pointed out rather dramatically how point set topology clarifies many concepts in modern analysis by visualizing the existence and uniqueness of the solution of an ordinary differential equation as a net of functions in the solution space with the compact-open topology that converges to the unique solution given the specified boundary conditions.

Of late, as I mature as a mathematician, I have begun to rethink the teaching of point set topology to mathematics students. There has been a very strong movement to dispense with teaching the subject altogether, as the semester-long traditional graduate course is being viewed as an anachronism that wastes the talented student’s time when he or she could get by perfectly well with the barest elements presented in the analysis sequence and the student could focus instead on much more research-relevant topological concepts such as homology and manifold theory.  This is neither the time nor the place to go into detail on my views. I will say I have some sympathy for this viewpoint: the traditional course on the subject rams a railroad car full of definitions, lemmas and theorems — most of which will never be used by the student again unless he or she chooses to specialize in analysis — down the students’ throats. But I think abandoning the subject to some limbo of outdated mathematics does a great disservice to future mathematicians. The concepts of point set topology arise in the most unexpected places and can be of great assistance in current research. Peter May and his coworkers are currently using point set topology to study the deep geometry of parametrized homotopy spaces, point set theory lies at the heart of dynamical systems and chaotic systems theory, and I myself am currently investigating in my research a class of set theoretic topologies on the integers identified by Kevin Broughan as the p-adic topologies.

Still, as I’ve said, there are very valid reasons to question and reassess how this material is taught to students. This is why Cunliffe’s book initially interested me. Any author writing a text on point-set topology is laboring under huge burdens. The first is the general negative attitude towards the subject outlined above. In addition, point set topology is one of the most saturated categories in the mathematics textbook marketplace. The prospective author stands to be crushed into obscurity under the sheer weight of the pile of topology texts.To have any chance of standing out from the pack, the author needs to offer something new.

Cunliffe claims the purpose of the book is to teach point-set topology to students with little or no background as an introduction to the language of pure mathematics. Since this is how I learned the subject back when I switched from biochemistry to pure mathematics, I was very interested to see how he’d do it. Such a book should be a kind of argument for teaching the subject in such a manner.

Alas, if that was his purpose in writing the book, he utterly failed.

First off, the book looks a lot cheaper then the $24.95 cover price. It’s issued by Bobo Strategy, one of those small publishing houses sprouting up all over the internet like dandelions. The cover is cheap laminated cardboard with some pictures of galaxies that look scanned off an old Carl Sagan pop book. Ordinarily, I wouldn’t care — the Dover paperback books aren’t exactly bound like a king’s private library either, and I like quite a few of them immensely.

The problem is what’s printed on those covers reads like someone slapped their old lecture notes — not even very good lecture notes — onto a scanner, and had a friend bind a couple of hundred copies for sale to an unwitting public. It covers all the usual topics in such a course in a very concise, definition-theorem-example format. One of the few creative moves is in Chapter IV, which includes sections on retractions and the fixed point property in topological spaces. These are certainly important topics that usually aren’t covered in undergraduate point set topology courses. Sadly, they’re not really covered here, either — Cunliffe spends a grand total of six pages on them. The point of his covering them at all is shown in the next section, on connectedness, in which he proves any topological space that has the fixed point property is connected. Such informative examples are very few in this book.

Peppered throughout are exercises — ridiculously few of them and most of them are easy enough for a brain dead baboon. There are some harder problems, but alas, nothing that could really challenge even a beginning mathematics student.

In short, there’s nothing here any student or teacher can’t find in any of the usual texts on the subject. Students should save their money for the train and what little food they can afford, and get Seymour Lipschutz’ wonderful Schaum’s Outline In General Topology — it covers everything in Cunliffe’s book and more, and does. Even better, you can have Allen Hatcher’s very nice lecture notes for free, or buy them in nicely printed form from Cambridge. Trust me on this, guys — you’ll be glad you did.

Andrew Locascio is finishing up his master’s work at Queens College Of The City University Of New York and the CUNY Graduate Center and is hoping he doesn’t get a heart attack before getting admitted to a PhD program. Between the stress and his weight which he can’t seem to lose, this may be a futile hope. He enjoys comic books, ranting, science fiction, ranting, mathematics, chemistry, physics, baking, good food and the company of tall beautiful women. On the last subject, being a less then an attractive man is somewhat of an advantage since he fades into the shadows to observe their character or lack thereof in social situations before deciding if they are worth his precious time to approach. He cried the day George Carlin died — he may never laugh again as Carlin’s brilliance was one of the rare experiences that truly made him do so. His opinions on these and myriad other things can be found at

Date Received: 
Saturday, August 16, 2008
Include In BLL Rating: 
Chris Cunliffe
Publication Date: 
Andrew Locascio

I Preliminary Material
1 Logic
2 Sets
3 Functions
4 Exam

II Definition of Topology
1 Topology
2 Standard Topology on R

3 Cofinite Topology
4 Closed Sets
5 Some Useful Tools
6 Some Notes on the Definition of a Topology

III Homeomorphisms
1 Basis for a Topology
2 Continuous Functions
3 Homeomorphisms

IV More Properties
1 Retractions
2 Fixed Point Properties
3 Connectedness

V More Examples
1 Subspace Topology
2 Lower Limit Topology, K-Topology
3 Sierpinski Space
4 Path Connected

VI Separation Axioms
1 T1
2 T2 — Hausdorff Spaces
3 T3 — Regular Spaces
4 Compact Spaces
5 Metric Spaces

Publish Book: 
Modify Date: 
Tuesday, October 28, 2008