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General Topology (Dover Books on Mathematics), by Stephen Willard
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Among the best available reference introductions to general topology, this volume is appropriate for advanced undergraduate and beginning graduate students. Its treatment encompasses two broad areas of topology: "continuous topology," represented by sections on convergence, compactness, metrization and complete metric spaces, uniform spaces, and function spaces; and "geometric topology," covered by nine sections on connectivity properties, topological characterization theorems, and homotopy theory. Many standard spaces are introduced in the related problems that accompany each section (340 exercises in all). The text's value as a reference work is enhanced by a collection of historical notes, a bibliography, and index. 1970 edition. 27 figures.
- Sales Rank: #240806 in Books
- Published on: 2004-02-27
- Released on: 2004-02-27
- Original language: English
- Number of items: 1
- Dimensions: 9.21" h x .80" w x 6.14" l, 1.06 pounds
- Binding: Paperback
- 384 pages
Most helpful customer reviews
44 of 46 people found the following review helpful.
Willard's General Topology - a must for every bookshelf
By Sheffielder
One of the purest and most intellectually challenging branches of modern mathematics, general topology is not a subject for the faint hearted. So it was a pleasure when I first encountered one of the best reference introductions to the subject to have seen the light of day. Willard's book remains one of my all-time favourites. It covers everything the aspiring topologist needs to know, and certainly supplies more than enough information for a potential PhD student to choose their initial area of specialisation. The chapters are split intelligently into sub-topics which move at a sensible pace from its introductory notes on essential set theory, through subspaces, products, compactness, separation and countability axioms, compactifications, and function spaces. Many of the "standard spaces" of general topology are introduced and examined in the large number of related problems accompanying each section. And for those wanting a bit more context than a maths book normally provides there's a detailed collection of historical notes for each chapter.
0 of 0 people found the following review helpful.
Best way to learn General Topology
By Alan Ottenstein
Fantastic book, it was the book for my three person presentation-based General Topology course, in which we basically had to do all of our learning from the book, and this book was very easy to learn from. It obviously takes effort and thought to read through everything, but I left every section with a thorough understanding of the topic. There are proofs for all major results, but they leave out the gritty details that you may want to go through on your own, a feature I liked. I can't imagine a better book to use to learn General Topology, or really any subject, on your own than this. From now on when I look for a good book to try to learn something independently, I will look for "the one most like Willard."
7 of 7 people found the following review helpful.
Absolutely amazing!
By Rodrigo Barbosa
This is certainly one of the best books on general topology available. It requires more maturity from the reader than the usual Munkres/Armstrong standard, but IMHO it is perfectly adequate for a first contact with the subject. It is a dense book, and it does not talk much like other books, but the exposition is so clear that this is actually a quality. Being succint, it manages to cover a lot more ground than the standard references; there is much more here than a one-semester course can cover. The exercises are usually difficult; some of them are real challenges (e.g. can you find an order in which the real numbers are well-ordered? This question pops out in the first set of exercises). The exercises are actually the purpose why this book leaves its rivals far behind. They provide the reader with a deep topological way of thinking in many ways: by forcing the reader to construct counterexamples himself (an essential skill for a topologist) and generalizing the theorems presented in the text, often to explore a new technique or construction. Sometimes this may provide the reader with multiple ways to look at a particular problem, which is certainly an useful skill (not to say inspiring!). A good example is the way the author explores the interconnection between nets and filters, which provide two different frameworks for describing topologies by means of convergence. Most other books describe just one approach or the other, and even when they do both they seldom explicit how they are related. A careful reader who works throughout the whole text, or at least through most of it, will have a better understanding of topology than the reader of the more usual texts. For the sake of comparison, I should say I found the discussion here about quotient spaces far clearer than Munkres's. Willard makes clear from the beggining the distinction between the "quotient approach" and the more intuitive "identification approach", which is the formalization of the intuitive grasp of cutting and pasting spaces. The author carefully develops both points of view, to show in the end they are really the same (in the sense of an universal property - i.e., up to homeomorphism). It becomes absolutely clear then that the first, more abstract approach, gives an effective way for manipulating mathematically problems arising in the second, hence its not-so-obvious-at-a-first-glance importance.
Readers who are already familiar with the methods and results of general topology and basic algebraic topology will also benefit from this book, specially from the exercises. This, together with "Counterexamples in Topology", by Steen and Seebach, form the best duo for studying general topology for real; this is the best option available for the ambitious student and the aspiring topologist. Also, as they are both Dover, the prices are ridiculously low. For a couple of bucks you may have access to some of the most beautiful treasures of mathematics.
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