Topology (Featured Titles for Topology) Hardcover – 28 Dec 1999
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Top customer reviews
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 (at least to the harder problems), so as to make the text more helpful for those readers who use it for self-study. Also a reviewer has correctly mentioned here 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, for a full-fledged coverage of the differential treatment. 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. This title is closely modeled in exposition and selection of topics on Munkres, thus nicely serving as a prerequisite.
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This text is excellent for self study assuming you've taken an analysis course and followed the proofs enough to do reasonably well in the exercises (when you screwed up-you figured out why.). General or point set topology is essentially math analysis distilled to its basic constructs and arguments (proof forms). Great theorems in analysis become great ideas in general topology. One theorem I've oft repeated is that a metric space is compact if and only if every infinite sequence (in it) has a limit point(in it)-or point of accumulation-this theorem is the prototype for the notion of sequential compactness. You'll see arguments from analysis repeated or called upon throughout-same friends just different clothes on them.
Even though this is an introduction I still look up proofs in it for things like the Tietze extension theorem, the Stone–Čech compactification, and the compact-open topology. A book at one level higher, which has material not contained in Munkres, is Willard, General Topology (Dover Books on Mathematics). An example of a theorem that is proved in Willard but not Munkres is that a product of *continuum* many Hausdorff spaces each with at least two points is separable if and only if each factor is separable (Theorem 16.4 in Willard). Willard is also better for the topology of function spaces. But Munkres is much easier to learn from and Munkres should always be used rather than Willard for a first course.
Additionally, I found the problems just a joy to work. They were very good at developing and then building understanding.
As a textbook, Munkres is clear and precise. He clearly states definitions and theorems, and provides enough examples to get a feel for their usage. The exercises are varied, but none were excessively hard, and they provide a good foundation to understand the flavor of topology. The prose is also very crisp and clear, and it provides motivation without had-holding and there is no needless obfuscation or verbosity. Having looked at many topology texts over the years, this is undoubtedly my favorite as a text. I would venture to say that this is the best introductory topology book yet written.
As a reference, Mukres is still great. It isn't as great a reference as it is a textbook, but it is still wonderful. The book's organization and clarity, which aids its function as a textbook, serves the reference user well. Additionally, it is fairly comprehensive insofar as basic point-set and algebraic topology are concerned. My one problem with Munkres as a reference: it is severely lacking with respect to manifolds and differential topology, even in their most basic form. Still, it is so wonderfully clear with respect to basic point-set and algebraic topology that I can't imagine wanting another book to fill in reference for those basic areas.
Seriously, this is THE book to learn topology, and then it should be kept around as a reference.
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