James R. Munkres

This is one of those obtuse books where even when you already know everything it is going to say, the presentation is turgid and confusing. The writing is tedious and fails to communicate the interest of topology either from an analytic or a set-theoretic point of view. You could read this whole book three times and still not know why topology was interesting or what it was used for.

Also, the binding is cheap and falls apart easily.

Recommended alternative: General Topology, by Kelley.

This book does an excellent job covering its subject matter in great detail. Munkres does not fudge on a single proof; there are no carelessly thrown about phrases that rely on the reader possessing a strong background in the subject (unlike some other math texts I own), and his explanations are as clear as one could hope. Although some might not like his insistence on using the formality of the theory throughout, I greatly appreciate the rigorous and at the same time insightful exposition.

Among the best mathematical texts I've ever read... really, fun to read.

An excellent introduction to point-set and light algebraic topology. If this is your first exposure to topology, I would recommend Kinsey's "Topology of Surfaces" as a companion of solid applications in the specific case of compact 2-dimensional topology.