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Topology: A First Course by: James R. Munkres

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Rating: - Would be better iff....

I am a grad student and wow, this is unbelievable. Some of the notation is inconsistent and all the problems are difficult. I am taking the class during the summer and it is a 24-7 ordeal.

Rating: - Excellent introduction: makes point set topology fun

This was my first introduction to point set topology as an undergraduate, and I enjoyed reading it even before taking the course.

Although not a hot research topic (compared to the rest of topology), it is foundational and as such many have assumed that point set topology could only be presented as a dull prerequisite for more interesting mathematics. Munkres' book, though, treats it as a goal of itself, as a fun world to play in, and as such, has attracted many students to topology.

It is recommended that a student first learn about metric spaces in a first-year undergraduate analysis class before learning about point set topology. Although the material is self-contained, the motivations for the definitions are hard to understand without knowing the more mundane examples.

Rating: - One of the best math textbooks I know

This is probably the best textbook on point-set topology (or general topology) ever written. Munkres is an excellent expositor. The book does demand a certain maturity; the definitions of a topology, a compact set, and a continuous function are quite unintuitive, and Munkres gives only a limited amount of motivation for them. Students with no experience with topological concepts in the context of, say, metric spaces will likely get lost quickly. But the more difficult theorems (e.g., Urysohn's Lemma, the Tychonoff theorem, and the Jordan curve theorem) are explained and proved very carefully in a "student-friendly" way. The book is also great as a reference, although some basic topics of importance to analysts are skimped on or omitted (Kelley's book "General Topology" will most likely have anything you can't find in Munkres). This book does not really discuss algebraic or geometric topology (besides a discussion of the fundamental group and covering spaces), which for most people are the really interesting parts of topology. Luckily, Munkres has written another book, "Elements of Algebraic Topology," which at least partially meets that need.

Rating: - highly recommended

If you are teaching yourself topology, this is the only book to use.
Prof. Munkres has a gift for explaining the concepts and proofs. After you pick up the basics, you should look at Counterexamples in Topology by Steen and Seebach.

Rating: - A standard introductory topology book.

Topology is a very beautiful subject, as many mathematicians will tell you. Point-set topology, the material which makes up the first five chapters of this book, however, is closely related to real analysis, and thus (in my humble opinion) quite dull. Nevertheless, Munkres does manage to make the study of point set topology bearable, and, in retrospect, possibly fun; the exercises consist of puzzles which are quite pleasant, if sometimes excruciating. Munkres treatment of algebraic topology is cursory and does not do justice to the subject; Massey's book would be a better introduction. This is a standard text, but make sure that you buy a simpler one if you are studying by yourself.