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General Topology (Dover Books on Mathematics)

20 customer reviews
ISBN-13: 978-0486434797
ISBN-10: 0486434796
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Product Details

  • Series: Dover Books on Mathematics
  • Paperback: 384 pages
  • Publisher: Dover Publications (February 27, 2004)
  • Language: English
  • ISBN-10: 0486434796
  • ISBN-13: 978-0486434797
  • Product Dimensions: 6.1 x 0.8 x 9.2 inches
  • Shipping Weight: 13.6 ounces (View shipping rates and policies)
  • Average Customer Review: 4.6 out of 5 stars  See all reviews (20 customer reviews)
  • Amazon Best Sellers Rank: #274,769 in Books (See Top 100 in Books)

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42 of 43 people found the following review helpful By Sheffielder on March 25, 2000
Format: Hardcover
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.
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25 of 25 people found the following review helpful By Johan Nystrom on February 14, 2008
Format: Paperback
First a caveate: This book may not be the most suitable for everyone that takes a FIRST course on General Topology unless he or she is prepared to put in quite a lot of work. This is because the book contains so much information in relatively few pages that the material is necessarily quite dense. Even so the book is a good purchase because it's cheap and will serve everyone good later as a reference.

The organization of the book: Everything is presented in a perfectly logical order, beginning with a summary of Set Theory and ending with topologies on Function Spaces. During the course the reader is invited to make excursions to other areas of mathematics from a topological point of view and perhaps gain insights into those fields that even specialists don't have. This is mostly done through problems for the reader to solve.

Definitions and Theorems: The definitions are always the most general possible, often presented as a set of axioms that the defined quantity has to fulfill. The theorems are almost always presented in their most general form.

The Proofs: The proofs are generally on either the shortest and most elegant form possible, or taken from the original publications. This is for the benefit of the reader even though it might appear to some readers as "terse" proofs because this kind of proofs is the one that gains the reader the most insight once they are understood. "Short and elegant" does NOT mean that the author leaves out details (unless they are explicitely assigned as problems).

Explanations and Motivations: The text is short and to the point. This again does not mean that the author leaves out anything relevant or that he does not warn for possible pitfalls.
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20 of 22 people found the following review helpful By another reader on July 28, 2005
Format: Paperback
I have yet to see Dugundji's topology text (it's always checked out at my university library) but I would still guess that this is one of the best there is. Willard's book most certainly covers all the topics that "every young analyst should know", as Kelley wrote, but I think this book outdoes Kelley's when it comes to that! It covers topics like convergence, separation & countability, compactness, connectedness, uniform spaces & a short discussion on function spaces & C* algebras at the end. Many of the theorems, proofs & examples come directly from the original source articles, or are the most general versions there is. Each section has a very detailed & interesting historical discussion at the end of the book where the author lists the original articles & the circumstances in which they were published & other stuff. Where the text really stands out though is the problem sets. As the reader goes deeper into the book of course the concepts get more complicated & proofs of extremely deep & important results are outlined as problems. These are things such as the Cantor-Bernstein theorem, Hahn-Banach theorem, Pontryagin duality theorem, stuff about realcompactness, Edwin Hewitt's construction of a regular T1 space in which every continuous function is constant(!) & many more. Don't be afraid though; the discussion in the text, hints given & notes in the back help with proving things like those. Many of the examples are also highly nontrivial & therefore very helpful (imho). To sum up, I believe this book is very underrated & deserves the recognition of the texts by Munkres & Kelley.
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10 of 10 people found the following review helpful By ikantspel on June 18, 2008
Format: Paperback Verified Purchase
Willard's text is a great introduction to the subject, suitable for use in a graduate course. I am personally not training to be a topologist but I must say that I enjoyed this book thoroughly and walked away with a firmer appreciation of the subject than I had previously had.

There is quite a bit of content ranging from subject matter and an extensive bibliography to a collection of historical notes. The exercises are suitable and doable; I have personally found that most of them range from being easy to moderately challenging but there are plenty of difficult problems as well.

It is important to note, however, that this text is primarily focused on point-set topology. There is a brief exposition of homotopy theory and the fundamental group but nothing compared to, say Munkres. But this is by no means a drawback. Willard thoroughly examines many topics that Munkres sometimes allocates to the exercises. A good example of this is net convergence, a topic that in my opinion, ought to be treated in any introductory topology course. In fact, Willard's development of nets makes for a nice, quick proof of the Tychonoff Theorem while Munkres's approach necessitates the development of a few technical lemmas.

Overall, this book is quite pleasant to read. It is also quite pleasant to purchase compared to several other introductory texts that run anywhere from 50.00-100.00. There are many nontrivial aspects to topology and this book has a way of gently nudging the reader into some of the more technical and delicate aspects of the theory. But as I mentioned before, while this book is a great introduction to point-set topology, this is not the text to read if one is searching for an introduction to algebraic or differential topology. In the latter case, Munkres or Fulton would be a good bet.
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