Topology (2nd Edition)

This introduction to topology provides separate, in-depth coverage of both general topology and algebraic topology. Includes many examples and figures. GENERAL TOPOLOGY. Set Theory and Logic. Topological Spaces and Continuous Functions. Connectedness and Compactness. Countability and Separation Axioms. The Tychonoff Theorem. Metrization Theorems and paracompactness. Complete Metric Spaces and Function Spaces. Baire Spaces and Dimension Theory. ALGEBRAIC TOPOLOGY. The Fundamental Group. Separation Theorems. The Seifert-van Kampen Theorem. Classification of Surfaces. Classification of Covering Spaces. Applications to Group Theory. For anyone needing a basic, thorough, introduction to general and algebraic topology and its applications.

The book is divided into two sections, the first covering general, i.e. point-set, topology and the second covering algebraic topology. Exercises (without solutions) are provided throughout. The exercises include straight-forward applications of theorems and definitions, proofs, counter-examples, and more challenging problems.

My only complaint with this book is that it does not discuss manifolds and differentiable topology, but other texts fill this gap.

I highly recommend this book to anyone interested in studying topology; it is especially well-suited for self study.


The best rigorous introduction to general topology!   January 5, 2005
 53 out of 56 found this review helpful

I used to own the 1975 (first) edition of this title since the late 1990s, but quite recently purchased the new edition as well, and donated the old book to our campus library. Before anything else, let me express that from the many topology texts that I have come across over the years, this one easily stands out as the best rigorous introduction for a beginning graduate student. It covers all the standard material for a first course in general topology starting with a full chapter on set theory, and now in the second edition includes a rather extensive treatment of the elemantary algebraic topology. The style of writing is student-friendly, the topics are nicely motivated, (counter-)examples are given where they were needed, many diagrams provided, the chapter exercises relevant with the correct degree of difficulty, and there are virtually no typos. The 2nd edition fine-tunes the exposition throughout, including a better paragraph formatting of the material and also greatly expands on the treatment of algebraic topology, making up for 14 total chapters (as opposed to eight in the first edition). I particularly found useful the discussion of the separation axioms and metrization theorems in the first part, and the classification of surfaces and covering spaces in the second part. In my opinion, after going through the discussion of algebraic topology in Munkres, the students should be ready to move forward to a (now standard) text such as Hatcher, for further coverage of homotopy, homology and cohomology theories of spaces. Eventhough a few contending general topology texts --such as a recent title published in the Walter Rudin Series-- have started to hit the academic markets, Munkres will no doubt remain as the classic, tried-&-trusted source of learning and reference for generations of mathematics students.

The one thing that should be mentioned though, one would wish there were some more hints and answers provided, at the back of the book, so as to make the text more helpful for those readers who use it for self-study. Before I finish, let me note that a reviewer here has correctly mentioned that Dr. Munkres does not include differential topology in his presentation. I speculate this is perhaps because he has already written a separate monograph on the topic. In fact, it is also necessary to get a handle on some fair amount of algebraic topology first (such as the notions of homotopy, fundamental groups and covering spaces), for a full-fledged treatment of the differential aspect. Regardless, one great reference for a rigorous and worthwhile excursion into the area (covering brief introductions to the Morse and cobordism theories as well), is the excellent title by Morris W. Hirsch, which is available on the Springer-Verlag GTM series. I would also like to mention that one other very decent book on general topology, which has unfortunately been out of print for quite some time, is a treatise by "James Dugundji" (Prentice Hall, 1965). The latter would nicely complement Munkres, as for example, Dugundji discusses ultrafilters and some more of the analytical directions of the subject. It's a real pity that The Dover Publications in particular, has not yet published this gem in the form of one of their paperbacks. The undergrad students testing the waters for the first time, should try Fred H. Croom's text, originally published in 1989 but now again re-issued and available in limited numbers and/or special order, through The Thomson Learning, Singapore. This title is closely modeled in exposition and selection of topics on Munkres, thus nicely serving as a prerequisite.


One of the best math textbooks I know   June 4, 1998
 22 out of 22 found this review helpful

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.


The best place to begin studying topology   August 10, 2002
 9 out of 9 found this review helpful

Although both parts of this book are exceptionally well written,
I've seen even better presentations of general topology in Sutherland's "Introduction to Topological and Metric Spaces", although admittedly Chapters 5 and 8 are not covered there. On the other hand I have found it very difficult to find a better book that covers part 2 of this book, Algebraic Topology. Most textbooks in this area either seem outdated or overly abstract. However, Munkres takes the time to explain concepts like covering spaces and the fundamental group with care and detail, providing a number of concrete examples. Combine this book with his differential topology book, and one can easily self-study his or her way to a mastery of first-year graduate topology.