Looking for some math textbooks

Could someone suggest a good textbook on set theory? I’m working my way through Halmos’ Naive Set Theory, but I’d like to have a second one (I don’t like learning from a single textbook.)

Also, can someone suggest a good book on surface topology? I asked for a topology textbook earlier, and Gemignani’s Elementary Topology was suggested, but I think I’m looking more for surface topology. (3-hole donuts and “Euler characteristic” and all that.)

Also, I read a SciAm article on manifolds, and somewhere down the line I’d like to study the Poincare manifold and suchlike. What’s a good (preferably elementary) textbook for that kind of thing, and what kind of prerequisites would I need?

-Ben

Concerning textbooks on surface topology and manifolds: I honestly haven’t seen any books which match your description. My own education started with some very elementary books for “laypeople” (The Mathematical Tourist by Ivars Peterson and The Fourth Dimension by Rudy Rucker both have a little of what you’re looking for), and eventually I started to see books aimed at graduate students (Three-Dimensional Geometry and Topology by William Thurston, for example), but none of the stuff in between came in textbook format. It was all in the form of hand-written lecture notes at best.

Since then, I’ve heard of some books which might be what you’re looking for: The Shape of Space: How to Visualize Surfaces and Three-Dimensional Manifolds by Jeff Weeks, for example, and something by Stephan Carlson called Topology of Surfaces, Knots, and Manifolds: A First Undergraduate Course. But I haven’t read those myself, so I can’t give much of an informed recommendation.

Once I’ve bumped this, hopefully one of the other topologists will wander by with some better info. (Where’s Cabbage?)

As far as prerequisites: definitely go through those books on set theory and elementary topology that you’ve already got. The set theory books will teach you most of the notation, and the elementary topology books will help with the terminology.

Elliott Mendelson’s Introduction To Mathematical Logic, 4th. Ed. has a big chapter dedicated to set theory. Be forewarned; this book is decidedly non-trivial to read and understand. As a result, you will probably need to read most of the first two chapters to understand the set theory.

And as long as were on the topic, I’m interested in elementary topology books (preferably the cheap ones from Dover, but I’m flexible).

Thanks guys, I’ll check out your suggestions.

As for elementary topology books, Dover has “Experiments in Topology,” and IIRC “Elementary concepts of topology” (the latter of which didn’t look very elementary, from a quick browse.) There’s also a book entitled “Visual Topology” which goes over graph theory and knot theory in addition to rubber-sheet topology. I haven’t had a chance to read these yet, so I don’t know how good they are.

-Ben

I found Kaplansky’s Set Theory and Metric Spaces wonderful and so too it seems did the solitary reviewer at Amazon.