books added: general topology (point set topology)

Introduction

Let me discuss my favorite general topology, i.e. “point set topology”, books. I have already discussed “algebraic topology” here.

Like so much other pure mathematics that I do not use professionally (for modeling power plants), topology is not on the tip of my tongue. But it’s fun, so I do it once in a while. And it’s fundamental, so I often have to go back to it when I’m playing with other mathematics.

This is a discouraging review in one respect: 4 of these 10 books are out of print: Kasriel, Dugundji, Sieradski, and Seifert & Threlfall. Heck, if you want Seifert & Threlfall, you should buy it in German! And for two of the books (Naber, Chinn & Steenrod) that Amazon claims to have in stock, there are multiple listings, many of which say the books are not available.

But I’m not going to go buy more books just because the ones I have are out of print. This is what I like, of what I have.

Buying Used Books


Let me point out that I buy used books online via abebooks.com. I have no connection with them except as a mostly satisfied customer. I think I have bought 10-15 books from them, and only one was not as advertised: it did not have “clean unmarked pages”. OTOH, it was so rare and hard to come by that I chose to keep it anyway: the student who marked it up stopped after page 51 and the rest of the book, the good stuff, was clear. (I could have returned it at the bookstore’s expense with a full refund, but I chose not to.)

I also buy used books online from powells.com; it’s a huge physical complex of bookstores (it really is huge!) in Portland Oregon, and wandering through them is almost a religious experience for bibliophiles. I have no connection with them except as a completely satisfied customer. Abebooks includes books available from Powell’s, but I buy such books directly from Powell’s because I’ve been there and I like them.

Two First Courses

First, there are two fine textbooks, intended and quite suitable for, a first course. Munkres came highly recommended, and from my own reading of it, I would imagine that it is the standard text. Well-written, with lots of examples. Good enough that I bought everything else he wrote; fortunately for my wallet, that’s only 3 other books, one of which is already in the bibliography.

I have no idea how I got the other book, by Sieradski. It is used, worn but unmarked. I suspect I bought it in person, but I have no recollection of where. It’s a little denser than Munkres, by which I mean only that it seems to have fewer words between definitions and theorems, but it still has plenty of descriptive text, and it has lots of examples and drawings.

If I feel like reading topology, I grab both of these and see what catches my fancy.

Sieradski is out of print, but if you’re interested in topology and you ever see a copy, you should investigate it.

Three References

Then there are three classic texts, all of which I consider to be references, although two of them do have exercises, and hence they are in principle, textbooks. (In fact, both English-language books declare themselves to be both text and reference.)

The first is Kelley. Famed for its exercises (is that a good thing?), and like so many math books of its era, it has no drawings whatsoever. He said he thought of it as “What Every Young Analyst Should Know.” originally published in 1955, it was republished by Springer in 1975 and is still available.

The second is Dugundji. It was allegedly the textbook for my first-year graduate course, but the instructor didn’t use the book at all. To be honest, I don’t know that there even was a “textbook” for general topology back then, other than Kelley – which I bought at the end of the year from another grad student who was bailing on math. When I looked through Dugundji years later, I liked its organization; good enough to make up for its lack of handholding. It’s probably the first place I look to see whether something is true or false. It includes chapters on set theory, ordinals and cardinals. Oh, it has no drawings either.

If you know enough to be looking for reference books in topology, you probably know better than I whether you want to find a copy of Dugundji.

(I am not sure where or when I learned what I know of topology. it might very well have been the combination of Dugundji & Kelley. both are reasonable places to look if you’re tripping over topology in, say, functional analysis. That is, they don’t have a lot of motivation: just the math, ma’am. But when I brought my own motivation, they were fine.)

The third is Seifert & Threlfall, but I list it not because I know it, but for its language. No, not for its exposition, literally for its language. It is available only used, and last time I looked, the cheapest English translation was $300. The book is frequently referenced for its counterexamples, and it is highly praised in books that do more than just list references.

I found a German reprint by Chelsea, in excellent condition, for $35. Having recently read some Harry Potter in German, I think my high school German is good enough for a math book. Since it’s an order of magnitude cheaper, I encourage you to buy the German rather than the English, if you think you can handle the language.

Three Proto-topology

The next group is three books which spend a lot of time on proto-topology, as it were. The first is Kasriel. I love this book because it has, in one volume, and in order, the topology of the real line, of R^n, of metric spaces, and then it does general topology. No running for an advanced calculus book to compare and contrast a metric space theorem with the corresponding R^n theorem. It is typical that I love seeing topology and metric spaces in the context that justifies the generalizations. In contrast to the next two books, you could learn topology out of this book.

Kasriel essentially gives us the historical path to point-set topology.

Unfortunately, Kasriel is out of print. If it sounds interesting, either look for a used copy, or look for another book that has the same organization: lots of R^n, lots of metric spaces, and topology, in one binding.

The second is Naber. This book has a lot of applications of topology, all in R^n, without ever even defining “topological space”! As I like seeing the context of topology in Kasriel, I enjoy the Naber book as a chance to see topology applied. I would not have wanted this book before I knew general topology per se, which defeats its purpose, but I look forward to working thru it someday. It covers simplicial complexes, homology, homotopy, and vector fields, all within R^n.

Be careful when you look for it: the 1980 paperback is out of print, but the book was reprinted in 2000.

The third is an ambitious little book by Chinn & Steenrod. (If you know enough to wonder, yes, that Steenrod.) It’s a lot like the Naber book, trying to do topology without defining it per se. More to the point, this book has two parts: the first wishes to motivate and prove that a continuous real-valued function defined on a closed bounded interval has and attains its extrema, and that y = f(x) can be solved for each y in between those extrema; the second part moves to 2D, where just stating the theorem is an adventure. It has a much simpler goal than Naber, but it’s still a fairly sophisticated goal.

There are things in the second part that I’ve seen but cannot say I’ve understood: specifically, winding number of a curve and index of a vector field. (Did I miss that in Apostol II? Yes! But not by much: it’s only a page or two. I suppose I should be embarrassed but I’ve gotten used to not knowing everything.) Still, I need to work thru this book Real Soon Now. The truly terrifying thing about this book is that its intended audience was “high school students and laymen”. (Was it accessible to high students before the “new math”?)

Be even more careful: although Amazon says it has it in stock, as I write this, it has so many conflicting additional entries saying it’s not available, that I’m a little uncertain.

Two Others

Then, there is a little workbook by Adamson. Too sparse to be a text, but lordy, what a wonderful way to review! 70 pages of definitions and exercises, followed by 78 pages of proofs. You don’t have to hold a piece of paper over the proof to try working it out for yourself. (While checking to see if it’s in stock, I learn that he has also put out a workbook in set theory. hmm….)

Having said that, I must admit that the author intended this as first course, for “independent study based on material carefully prepared by a teacher who otherwise gives minimal assistance.” (If you’ve ever heard of R. L. Moore and his method, you understand.)

Finally, there is a Dover paperback by McCarty. I probably picked it up because it was a Dover, but I might have seen it in a list of references somewhere. It presents just the topology required to get to topological groups, but along the way it has a very nice introduction to commutative diagrams, and the entire book is extremely readable. I sat down to see what was in it, and didn’t put it down until I’d read the whole thing!

The Books Added

Adamson, Iain. A General Topology Workbook. Birkhäuser, 1996.
ISBN 0 8176 3844 X.
[general topology; 10 Nov 2008]
Upper division. From the introduction: “This book has grown from my attempts to provide a self-learning introduction to general topology for several generations of students….” Answers. Guide to further reading.

Chinn, W.G. and Steenrod, First Concepts of Topology. L.W. Singer & Random House1966.
[general topology, algebraic topology; 10 Nov 2008]
From the introduction: “… to show how topology arose, develop a few of its elements, and present some of its simpler applications…. Our presentation … will be centered around two existence theorems… {in one and two dimensions respectively}.” Answers. Guide to further reading (3 books). Epilog.

Dugundji, James. Topology. Allyn & Bacon, 1966.
[general topology; 10 Nov 2008]
Out of print. Reference and text. If they reprint it, i’ll call it a classic. Some homotopy theory.

Kasriel, Robert H. Undergraduate Topology. Krieger, 1977.
ISBN 0 88275 444 0.
[general topology, metric spaces, euclidean spaces; 10 Nov 2008]
Out of print. From R and R^n to metric spaces and then to topology. From the preface: “… it is to {the graduate student’s} advantage to have taken a course in general topology before beginning his graduate program…. essentially self-contained except for elementary calculus.”

Kelley, J. L. General Topology. Springer, 1975 (was van Nostrand, 1955).
[general topology; 10 Nov 2008]
Graduate Text in Mathematics. Classic reference and text. From the preface: “… a systematic exposition of the part of general topology which has proven useful in several branches of mathematics.” Epilog and guide to further reading – in some of the exercises!

McCarty, George. Topology: An Introduction with Application to Topological Groups. Dover, 1988 (orig 1967).
ISBN 0 486 65633 0.
[general topology, topological groups; 10 Nov 2008]
Upper division. How can I not like a book which says, in an exercise, “… make the assumption that every subgroup in sight is closed.” A very readable book. Guide to further reading. Epilog. (Both at end of chapters.)

Munkres, James R. Topology. Prentice Hall, 2000 (2nd ed).
ISBN 0 13 181629 2.
[general topology, algebraic topology; 10 Nov 2008]
A very well written senior and first-year graduate textbook. Lots of words between the theorems provide lots of motivation. If this isn’t the premier textbook, I’d really like to know what is.

Naber, Gregory L. Topological Methods in Euclidean Spaces. Dover, 2000.
ISBN 0 486 41452 3.
[algebraic topology, differential topology, euclidean spaces; 10 Nov 2008]
From the preface: “… to persuade students… that the evolution of topology from analysis and geometry was natural and, indeed, inevitable; that the most fruitful concepts and most interesting problems in the subject are still drawn from independent branches of mathematics…. an ambituous agenda of topics has been included….” Guide to further reading. Answers.

Seifert, H. and Threlfall, W. Lehrbuch der Topologie. Chelsea, 1945.
[general topolgy; 10 Nov 2008]
Out of print. Classic reference. I bought it for the counterexamples; I bought it in German for the price.

Sieradski, Allan J. An Introduction to Topology & Homotopy. PWS-Kent, 1992.
ISBN 0 534 92960 5.
[general topology, algebraic topology; 10 Nov 2008]
Out of print. A very well written senior and first-year graduare textbook. From the preface: “Most topics are developed slowly in their historic manner, in order that a newcomer not be overwhelmed by the ultimate achievements of several generations of mathematicians.”

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