Thoughts on Henle's Topology book?

[proximal]

Hi all! I'd like to ask for some opinions on a book.

I'm currently taking an undergraduate course in topology. We're using the book A Combinatorial Introduction to Topology, by Michael Henle, and so far I have mixed feelings about it, feelings that my class and professor seem to share.

1. The book doesn't seem to cover enough point set topology for a proper introduction to topology. (Granted, the author seems to be going for a unique approach here.)

2. The book uses less familiar terminology. A point being "near" a set instead of being a "limit point" of a set, and so on.

3. The proofs in the book aren't always very clear.

4. The book treats only topology in the plane, and does not do much with higher dimensional Euclidian spaces, or abstract topological spaces. Some theorems are restricted to the plane, though they hold true in more general situations. (The Bolzano-Weierstrass theorem, for instance.)

5. The exercises are...weird, sometimes. There's the ever-present confusion of whether we're allowed to assume we're in the plane, since that's what the book focuses on. In one problem involving Euler characteristic the n-gon faces of a polytope were just referred to as "cells," causing some confusion.

My professor has resorted to addressing these issues by using some notes on point set topology from a different professor's webpage.

As I understand it, the math department at my school switched from Munkres to Henle fairly recently, the idea being that Munkres was too abstract and didn't include enough pictures for students to develop intuition. Henle has plenty of pictures, and a point in favor of Henle was that it would also introduce students to interesting phenomena in low-dimensional topology --- Mobius strips, Klein bottles, and so forth.

Maybe Munkres is indeed too abstract (I wouldn't know, I haven't read it yet), but I find that professors are often very capable of drawing pictures on a blackboard (mine could probably turn it into a profession with practice), and in fact that's what lecture is for, isn't it? There's also that the course is supposed to introduce the ideas of topology that are useful for analysis, and while I don't know much about algebraic topology it still seems weird that point set topology is so neglected in an introductory topology book.

Has anyone studied/taught/read from Henle's book, and could you tell me your opinions on it? How would you compare it to Armstrong's Basic Topology or Munkres's Topology?

 

[micromass]

That sounds like an absolute horrible book to study from. You would be much better off with a standard text such as Munkres (don't take Armstrong, that's a horrible book as well). I don't say Munkres is the best topology book out there, but it sounds so much better than the crap they've been teaching you right now.

 

[jbunniii]

If you're looking for a good, cheap introduction to point-set topology for supplementary reading, I highly recommend Mendelson's Introduction to Topology. It's crystal clear and, if you're in the US, Amazon sells it for only $5.78.

 

[proximal]

Sorry for the late reply, and thanks for the replies.

In the meantime I'm looking for a copy of Munkres, but just to clarify: I'm looking for someone who has read/used/taught from Henle. After all, there's got to be some merit to the book. A department doesn't switch from Munkres to Henle without a very good reason.

 

[rrbold]

Henle's book is a great book for a terminal course. We use it to teach future
high school teachers and others who will not be taking more advanced topology.

If you will be using topology in future work, the best intro to point set topology is
Willard's General Topology, now reprinted by Dover at about $16 on Amazon. It is
the most comprehensive, and is elegant. As each concept is introduced he provides
numerous alternative formulations and rapidly proves their equivalence. For example,
topologies are defined in 4 ways: first by open sets as usual, then by closed sets,
clearly equivalent, but it is worth stating the exact form the axioms for closed sets should
take, then by the closure operator, A |--> Closure(A), with its axioms, then by
the interior operator, and, since I'm doing this from memory, not consulting the book, I now realize I left out some others: bases, subbases, neighborhood systems. Numerous exercises allow the student to work out loads of examples and a lot of the details that are pretty standard. The voluminous exercises are one stellar feature of Willard. They provide a useful source of counterexamples, for one.

The definition of a topology is a minor example. Complete details for convergence via nets and equivalently via filters, are given. Inadequacy of sequences, unless the space has countable nhood bases. All the separation axioms you might want. All the varieties of compactness. Varieties of connectedness. Etcetera, etcetera. It really is heads and shoulders above the standard texts like Munkres, Mendelsohn, Armstrong, and Kelley (old but worth looking at too). It isn't chatty, and may be tough slogging for some, but worth the effort.

Now, re Henle. Someone complained about his def of continuity via nearness. Actually this is very pretty. Let x be near A if x is in the closure of A. Then f is continuous iff x near A implies f(x) near f(A). This is the cleanest and most intuitive def of continuity I can think of. Examining what it means in detail leads directly to the usual tests in terms of neighborhoods or open sets. As I mentioned in my discussion of Willard, defining topologies via the closure operator is perfectly equivalent to the usual definition.

I generally augment this section of Henle by discussing
- neighborhood systems,
- when the identity map id_X : X --> X between X with two different nhood systems is a homeomorphism,
- hence when two nhood systems define the same topology, and finally
- observing that the collection of all open sets is the unique maximal nhood system in each resulting equivalence class. (Please pardon my sloppy English here.)

I agree that his terminology is sometimes a bit silly. The instructor can easily compensate.

Virtues of Henle: with a minimum of prerequisites a student gets to see some of the big
theorems that would never be reached in a standard point set course. For an audience of people who are not going on to do courses in algebraic, geometric, or differential topology or to do research, this is far more important than seeing how the axioms work out.

Big theorems:
1. Brouwer fixed point theorem via Sperner's lemma, a beautiful, simple, combinatorial argument that easily generalizes to all dimensions.

2. Poincare Hopf Theorem, and along the way a nice intro to qualitative analysis of diff eqs, i.e., dynamical systems.

3. Jordan curve theorem by a very clever form of mod 2 homology. This is really the usual Mayer Vietoris sequence argument, but stripped down to essentials.

4. Combinatorial surfaces and their classification. I had always seen allusions to this as a student, but had never seen the arguments until i taught it. It is fun.

5. Homology groups, first mod 2 then integral. This is usually as far as I can get in a semester, so I cannot comment upon his last chapter.

It does require an instructor who is more than a few pages ahead of his students. Preferably, one who knows the more sophisticated versions of these theorems. If you look closely, you'll see that Henle's arguments are solid, but it may be hard if you're just reading it line by line. The instructor should just take Henle as an outline and fill in the arguments on her/his own.

Your mileage may vary, of course. Hope this helps.