General Topology (Dover Books on Mathematics)

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. Examples of introduced concepts and definitions: There are numerous well chosen examples, often nontrivial, to illustrate the meaning of introduced concepts. The problem set: The set of problems is just fantastic. The problems are numerous, diverse, illustrative, and again, sometimes HIGHLY nontrivial. Don't be too scared though, because the author provides very accurate hints of how to approach the more difficult ones. Bibliography, Historical Remarks and Index: One just has to admire the amount of work the author has put into this. Miscellaneous: As mentioned, the material is (necessarily) condensed, but the text is never "dry" or boring. There is an undertone of humour in quite a few places. For instance, when the author mentions that not every regular space is completely regular, because there exists a formidable example that shows this fact, he relegates that example to problem 18G "where most people won't be bothered with it". This practically guarantees that most people WILL be bothered by it by looking up 18.G. There, in 18G, he provides som many hints that it is actually doable for most people to reconstruct this formidable (i.e. difficult) example. On the Downside: There are no solved problems, and the author does not teach the reader on HOW to solve problems. This is however compensated for by the numerous hints in the problem set and through the methods of thaught one learns from reading and understanding the proofs. Also, in topology, one basically has to invent ones own mothod to solve an unsolved problem. There is no canonical way of doing things!

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.

This is a wonderful General Topology book that explores all aspects of General Topology in a deep way. The author doesn't hold back and provides you with all possible perspectives in a given topic. To this, the exercises add even more depth by providing you with interesting examples and counterexamples as well as letting you find some really importnant results for yourself before they are properly examined in the book. All of the above make this book an excellent introduction as well as reference for General Topology. A fair warning is that before reading this book you should have mastered Metric Spaces and their Topology, because they are used as motivation for a lot of definitions which otherwise seem unnatural at the first glance. Chapter 2 of Rudin or chapters 1 and 2 of Tao's Analysis 2 should suffice.

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.

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."

Hands down, this is one of the best book in the series of "Dover Books on Mathematics." For students that have no prior exposure to topology, try to read Mendelson's Introduction to Topology, also a Dover book.

Great book with a lot of good stuff, especially in the exercises. However, if you're not willing to work through the relatively dense exercises, you might want to look for something that reads more easily and come back to this book later.

Definitely the best introduction to point set topology available. Superior to Munkres if you can handle the higher level of abstraction in Willard.

tutto OK

If you are looking for the newer edition of this book, no need. It is exactly the same. If you are still adamant and want to confirm, i am sure you can download one from your University library.

Es mi libro de cabecera para topología general, tiene discusiones breves pero concisas y muchos apéndices donde se abordan temas avanzados. Me fue muy útil incluso a nivel posgrado.

The subject ain't easy and the motivation usually given in books lack motivation, but here the author gives a real motivation and all of this with the highest degree of rigor. Munkres book is not bad, but its too wordy sometimes and you can easily miss the point and is of course less advanced than this text. Nevertheless, this is an introductory text and should be available to a beginner on the subject. Just make sure you get a problem book on the subject since only doing the proofs and exercises will get you the knowledge (There is a "Solutions Manual" on the web for this, but in my opinion isn't enough.) A particularly good problem book is the one by Mohammed Hichem Mortad named "Introductory Topology : Exercises and Solutions".

After 25 years of teaching, I consider this one of the best explained, most comprehensive treatments of a subject I have come across. Definitely not a first book to understand metric and topological spaces, but for a more advanced treatment absolutely superb.