Old, terse and classic. If you want chatty and a lower level treatment then buy Munkres. This is a really good - but is not an easy - book.

This is far and away the best text I've used to learn topology. I'm currently working through (read: proving everything from scratch, doing all problems) chapter 10 of the text -- roughly a little over halfway -- and it's fantastic. Topology is a beautiful, beautiful subject, but no text (at least none that I've come across) have done the subject that right mix of intuition, rigor, and careful organization that Dugundji provides. For example, the standard text on the subject, Munkres, almost bored me to tears before I dropped it and moved on due to its sheer lack of motivation; Lee's book on topological manifolds does a little better, but is more specialized and moves at a rather excruciatingly slow pace.

Dugundji does a spectacular job of motivating most (though admittedly not all) of the topics covered in the text, and everything is presented in a very coherent order that allows the student to see how the topics are connected (the schematic on the last page of chapter 15 is wonderful). For example, his division of topics into axioms of separation, covering, compactness, etc. really help the student to conceptually organize the subject as a whole. Overall, I'd recommend the text any day over the standard books being used in the classroom, but as a word of warning, the book does require a substantial amount of mathematical maturity to fully appreciate.

One of the best Topology books I have read. Even though the book has no figures (as one would expect from a topology book), almost every detail is covered and there are not obscure parts in the proofs. For example, the book by Willard is also good, but in some parts there are more complex details left for the reader. I took a basic topology graduate level course on the first half of 2007, which consisted on solving the problems in this book. We were able to find some problems that asked to prove something false, but they were three or four among all the problems from sections III to VIII. Anyway, this book is a classic that you should own if you plan to work in topology or at least read it while studying the subject. It's just a shame that the book is out of print.

Good

There are only a few classic encyclopaedic texts on undergraduate topology, and Dugundji's is one of them. And among such books, this is my favourite because the others are too old-fashioned or too voluminous. Dugundji's book is short, modern, and impeccable. It covers every topic an undergraduate should know and even more. It is still useful for me after years of use. It exposes all important concepts of set topology and gives a short but focused introduction to algebraic topology.

You won't regret to read it.

Clear, consider, straight to the point, does not waste time in unnecessary commentary or topics that aren't very interesting, comprehensive, does not leave many details for the reader (which is a huge plus in my experience), exercises are doable. It is truly an insanity this isn't the standard book for a first course in topology.