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Customer Reviews (10)

agonizing to use as a reference
The back cover blurb describes this book as "among the best available reference introductions to general topology." Notice that word "reference." I find using this book as a reference to be incredibly painful. The problem is simply that it was written in 1970, when word processors didn't exist. Therefore whenever I look up anything in the index, I get something like "uniformity of compact convergence, 43.5, 43.6, 43.11, 43C." That is, there are no page numbers. 43.5 is a definition of the term. 43.6 is a theorem. 43.11 is another theorem. 43C is an exercise. To find any of these, I have to laboriously flip back and forth, searching for the desired decimal-numbered definition or theorem, or numbered and lettered exercise. Starting ca. 1985, there was no longer any excuse for producing a book without a proper index referring to page numbers; the word-processor would do it for you. Since ca. 2005, it's hard to see the utility of a book like this as a "reference" at all, because I can find a better treatment of any given topic on Wikipedia.

Absolutely amazing!
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.

A Great Beginning Text
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 theTychonoff 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.

A masterpiece
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!

Excellent
this is an amazing book. very wisely constructed with a lot of real content.
if i may ask for something more i would ask for an updated version, and solutions for problems. ... Read more

Product Description
With more than 30 million copies sold, Schaum's are the most popular study guide on the planet. Mathematics students around the world turn to this clear and complete guide to general topology for its through introduction to the subject, includingeasy-to-follow explanations of topology of the line and plane and topological spaces. With 650 fully solved problems and hundreds more to solve on your own (with answers supplied), this guide can help you spend less time studying while you make better grades! ... Read more

Customer Reviews (9)

A Genuine Masterpiece Endowed with Eternal Beauty
This book is so good, that one wonders if anybody can ever write a better book. The solved problems in this Schaum's Outline are extremely instructive , and the supplementary problems are equally good.Extremely well explained Theorems , and very clear discussion of concepts.Although other mathematics outlines in the Schaum's are also very good , nothing matches the quality of this particular Outline.If anybody desires to learn General Topology on their own , this is certainly the way to go .But be prepared to work very hard to solve the supplementary problems , as General Topology is in itself a hard subject.

Great Book
This product takes the down and dirty information of Point-set topology and feeds it to the sophomore college student. Its a good read for those out of their first year of Calculus and really want to prepare or know what's up. Its used as a supplemental text for my Honors Course, and it really helped me get used to the topological definitions and notations. Filled with solutions and worked out problems it builds essential skills required and expected of the Junior level student. It's great for engineering majors, so they won't freak out when they have to deal with upper level math. I give it a very strong recommendation. But, just remember its higher level material, so don't expect it to go down like a Calculus course; it needs time for mental percolation.

An excellent supplement for the learning of topology
When I was taking a master's level course in topology, the first three weeks were easy, a simple continuation of what I had had in set theory, logic and analysis. Then things executed a change in the negative direction. I was lost, puzzled by some of the expressions and the purpose of some of the theorems.
In an attempt to right my mathematical ship, I went to the bookstore and purchased a copy of this book. It was money well spent, after a weekend working through some of the problems, I understood the ideas behind the theorems and was able to solve the problems given on the take-home tests. I received an A in the class and some of that is due to the example problems in this book.

Admiral Topology
This is an old and good book.
It's so good that deserves be called "ADMIRAL TOPOLOGY".

General Topology
This is a good book. It gives all the prerequisite info in the first couple of chapters in a clear and easy to read, logical order. Good price, also. ... Read more

Product Description
This book is designed for the reader who wants to get a general view of the terminology of General Topology with minimal time and effort. The reader, whom we assume to have only a rudimentary knowledge of set theory, algebra and analysis, will be able to find what they want if they will properly use the index. However, this book contains very few proofs and the reader who wants to study more systematically will find sufficiently many references in the book.

Key features:

• More terms from General Topology than any other book ever published

• Short and informative articles

• Authors include the majority of top researchers in the field

• Extensive indexing of terms
... Read more

Customer Reviews (1)

A good and expensive reference
This, as it's name tells, is a book that you can use as a desk reference on general topology. There are no proofs and references to results are limited. Also, this book is way too expensive. I bought it because I have made my mind to being a topologist for the rest of my life. That means that you should buy this book if and only if your life is topology. I also think this book is for the college libraries to buy and have them for everybody at a university campus (or at least in the institute or school it belongs at). ... Read more

Product Description

This is the softcover reprint of the 1971 English translation of the first four chapters of Bourbaki’s Topologie Generale. It gives all basics of the subject, starting from definitions. Important classes of topological spaces are studied, and uniform structures are introduced and applied to topological groups. In addition, real numbers are constructed and their properties established.

Customer Reviews (2)

a solid introduction
this book is a bit denser than most other introductory general topology books. But it does quite exhaustive survey of important concepts pertaining to general topology.

Since Bourbaki series builds upon its previous materials, many set theoretic ideas and terminologies are used without explanations. So unless one does have access to their previous book "Theory of Sets" there will be some minor frustrations/annoyances when reading this book.

For the content, it starts with open set axioms for the topology like any other intro. topology text.
Then Bourbaki shows how the neighborhood system determines a unique topology on a set and conversely. Next topic covered is continuity and the initial and final topology induced by a family of mappings and defines subset, product, and quotient topology in terms of the these two natural constructions. After covering these topics Bourbaki covers quotient various quotient mapping and some useful criteria for determining when the map from quotient space to the codomain after the canonical decomposition of a map becomes homeomorphism.

Next topic covered is open and closed mapping along with equivalence relations being open or closed.

After discussing general continuity without any major restrictions on the topological spaces, Bourbaki then introduces typical restrictions; namely compactness, Hausdorff, and regular conditions.

Unlike many other major introductory topology books, Bourbaki does not talk about sequences nor nets in order to define compactness( quasi-compactness as Bourbaki calls it). Instead, he uses filters to define compactness. Using Zorn's lemma, existence of ultrafilter is shown and Tychnoff's theorem is proven using filter property in a very slick fashion.
Also, there is a short section on germs, although this is not used in the rest of this book in any significant ways.
Then, Bourbaki moves on to the topic of the limit and cluster(accumulation) point of a function of filtered space into a topological space and shows how the definition limit of a sequence or nets can be retrieved from a definition of limits of a function with respect to a filter.

After covering this necessary tool or terminology, Bourbaki then covers Hausdorff space and regular space. Extension of a continuous function of a dense subset into a regular space, by continuity is shown in a very slick fashion.After covering this he does the typical stuff associated with compactness, paracompactness, and connectedness. These three sections are very similar to other intro. topology text in its content but with terminology adjusted for use of filter in these concepts.

However, Bourbaki offers something you do not typically see in intro. topology text, in this section; proper mapping and inverse system.
Proper mapping is shown as an alternative criterion for determining compactness, and other use of proper mappings are illustrated.

Next section of this book is uniform space, which is a generalization of pseudo-metric spaces.
Here, Bourbaki shows how a notion of completeness can be generalized to the setting of uniform spaces and introduces notion of Cauchy filter.The major result of this section is the construction of Hausdorff completion of a uniform space. This construction is essentially same as the construction of real numbers from Cauchy sequences of rational numbers but Bourbaki maintains the vocabulary of Cauchy filter. Also, instead of working with equivalent classes of Cauchy filters(or sequences if you prefer), Bourbaki uses a system of representatives called minimal Cauchy filters.

Section 3 of this book, covers topological group. Using how a neighborhood systems determines a unique topology, he quickly determines criterion for existence of suitable topology such that this topology is compatible with the pre-existing algebraic structure; i.e. all the algebraic operations become continuous with this topology. Thus the completion stuff one might see in Lang's Algebra or in Atiyah's intro. commutative algebra will makes more sense after reading this section.
Then the usual stuff of completion of topological group, ring, field, module is shown using tools developed in previous two sections.Also, using inverse system he does a few approximation stuff, which one can skip without disrupting further reading.

Section 4 is the last section of this book, and Bourbaki finally talks about real number. Since he talked about completion of topological group, he defines real number as the Hausdorff completion of rational numbers considered as an additive topological group. After this consideration many results just fall out; such as rational line being dense in real, etc.

After this characterization supreme property of a bounded set of real number is proved using Archimedes' Axiom(which is proved also). Then the usual criterion of compactness and connectedness in real line is proved.Here the proof of these facts are not given in the standard way deriving contradiction using supreme property. So it is interesting to see how the previous materials are used to prove these well know facts.

Then monotone convergence of a function from directed set into a real number is discussed and its consequences are discussed;limsup, upper envelope of a family of continuous functions, etc. Also, upper and lower continuity is discussed and some familiar results are discussed in brief fashion.

Finally, Bourbaki talks about series of real number and standard facts such as Cauchy's convergence criterion, alternating series test, etc are given along with n-ary expansion of real numbers.

And this is where part 1 of this book ends.

My overall impression is that this book(just like other Bourbaki book) is very user friendly, in that it does each proof very carefully. However, due to its constant build of a long logical chains, you really cannot read this book like a typical textbook; meaning you cannot skip around and the entire book must be read in a linear fashion.

Also, the filter and uniform stuff is not typically covered in the introductory topology courses so to a novice this stuff might not be useful to your classwork(at least for the undergraduate or beginning graduate level). However, reading this book broadens your view on general topology for this book explains ideas behind the common concepts you encounter in other courses; such as use of filtration in a module to define a topology in an algebra course.

Anyway, it seems to me that the biggest disadvantage of reading Bourbaki is its inefficiency.Meaning, stuff you really wanna see is not discussed unless you read through first 200 or 300 pages of this wonderful book.And this is probably the main reason why Bourbaki is not used as a standard text anymore;not because categorical language is not used as some might argue. So to a student with not enough studying time, this book will not useful when it is needed.

the book that the experts study
i recently ordered "A GENERAL TOPOLOGY WORKBOOK" by Iain Adamson, as i very much wish to understand the mathematics of topology.as i would do with any other book, i started by reading the introduction and suggested readings.Mr. Adamson highly recommends the BOURBAKI series on topology as a reference material.he ascribes to this series as his text of choice and further states that this is the text that he has studied the most closely.it is for this reason that i am ordering this series.

i extend my thanks to mr. adamson for the recommendation.with the plethora of choices in study materials and a limited budget, i needed to narrow my scope and decide which text would best serve me.

thanks again:-) ... Read more

Product Description
General Topology is not only a textbook, it is also an invaluable reference work for all mathematicians working the field of analysis. It has long been out of print, but a whole generation of mathematicians, including myself, learned their topology from this book. There are no wasted words in Kelley?s presentation; every sentence is short and to the point, but the student would do well to contemplate each of them, for they are pregnant with subtle implications. The numerous problems that follow each chapter are well chosen to complete the students? understanding of the topics discussed.THIS VOLUME gives a systematic exposition of the part of general topology which has proven useful in several branches of mathematics and is intended especially as a background for modern analysis.One of the many features of this volume is the wealth and diversity of problem material which includes counter-examples and numerous applications of general topology to different fields. The appendix, which is entirely independent of the rest of the book, includes an axiomatic treatment of set theory.The author has included the most commonly used terminology, and all terms are listed in the index. As a reference, this book offers a unique coverage of topology. ... Read more

Customer Reviews (6)

a splendid technical book
I was motivated to read this book while in grad school, becasue I needed to understand the French literature in my field (probability). One particular concern is the metrizability of a general topological space. I would say Kelley's book has a spendid presentation on this subject.

Other things in this book are also practically useful. Convergence in the general sense (net or filter) is useful in mathematical finance. The part on locally compactness and paracompactness is a must for anyone working in differential geometry. And if you work in analysis, then the chapter on space of continuous functions is a good reference to look up.

The exercise problems are also good resources when you need some help. I still remember one cute problem on the neighbourhood systems. It helped me understand how a family of seminorms would yield a topology on a linear space.

Evetually, I read this book from cover to cover. And I would say this is one of the best education I've ever received.

If there has to be a complain, the proofs are somewhat hard to read. But this is more or less determined by the nature of the subjects. And when you are well-motivated and equipped with certain mathematical maturity, this problem will gradually go off.

In summary, this book is comprehensive, useful and beautifully written. It is a treasure that every mathematician's library should have.

Generally great; a few annoyances
This is a great book.The proofs are clearly presented, and generally it is easy to understand the motivation behind definitions and theorems.Exercises are relevant, interesting, and well designed, often allowing the reader to discover things that other texts describe in dull detail.Unfortunately, a few exercises (such as "Integration Theory: Junior Grade") seem to pop out of nowhere.I consider this a minor defect.A much larger annoyance is that Kelley defines partial and linear orders in an utterly non-standard and somewhat clumsy way, which ends up affecting a large number of exercises.If you already know something about orderings, you will encounter many surprises; if you know nothing about them, you may get the wrong idea.

Topology with the analyst in mind!
I don't hesitate to give this book 5 stars. It is solid! Many reviewers allow too much personal judgement to cloud their appraisal of a certain book. To me I believe it is important to be as dispassionate as possible so that a prospective buyer can make an unbiased decision. Rather than label a book as "bad" or "good" one should focus on some factors such as:
(1) Content: a summary of the main point covered by the book (this is optional). In the case of this book, this is obvious from the title.
(2) the author's approach: Kelly took what I call the "analyst's approach" to topology. This is fine for those who love analysis but don't really care for topology for it's own sake (like me!) By using this approach, those like me are much more inclined to find topology motivating because ones sees it as abstractions of what one is familiar with
(3) the presentation: Kelly gave a simple but "sophisticated" presentation. You will not describe him as very expository but the presentation is excellent. Some people seem to prefer this style and some don't. No, this has nothing to do with the so-called "mathematical maturity" (how do you define that by the way?) What the author expects you to know to understand the book - that is, the intended audience - is usually stated clearly in the preface

May have been good in its day
I cannot agree with the other reviewers on this. Back in the days when there were hardly any general texts on topology this may have been good. Nowadays there are at least a dozen such that are far better than this. The printing fonts and layout are spidery and primitive and not easy on the eye. The style is rather formal and dry for a subject as rich as this and little effort is put into illustrating the material with background, diagrams or examples. As I said before there is no shortage of better texts amongst which Hocking & Young is worth special mention.

The great classic of point set topology
John Kelley wanted the title to be "What every young analyst should know", but was convinced (by Halmos, among others) not to use it. Still, it is a very good description of the book. Barry Simon calls it"superb" and recommends that you read it by trying to do theexercises,recurring to the text as needed. But then you would perhaps notpay attention to how wonderful the text is. I believe this is thebest-written modern mathematical text. The proofs are clean and extremelyelegant. The prose itself is beautiful and frequently witty. Treatstopological and uniform spaces at depth and in detail, so as to be both atextbook and a reference. Excels in both capacities. This is mathematicsclose to poetry. ... Read more

Product Description
Topology, the foundation of modern analysis, arose historically as a way to organize ideas like compactness and connectedness which had emerged from analysis. Similarly, recent work in dynamical systems theory has both highlighted certain topics in the pre-existing subject of topological dynamics (such as the construction of Lyapunov functions and various notions of stability) and also generated new concepts and results (such as attractors, chain recurrence, and basic sets). This book collects these results, both old and new, and organizes them into a natural foundation for all aspects of dynamical systems theory. No existing book is comparable in content or scope. Requiring background in point-set topology and some degree of ``mathematical sophistication'', Akin's book serves as an excellent textbook for a graduate course in dynamical systems theory. In addition, Akin's reorganization of previously scattered results makes this book of interest to mathematicians and other researchers who use dynamical systems in their work. ... Read more

Product Description
An undergraduate introduction to the fundamentals of topology — engagingly written, filled with helpful insights, complete with many stimulating and imaginative exercises to help students develop a solid grasp of the subject.
... Read more

Customer Reviews (13)

Awesome book!
For all those who want to get into the field of Topology and then do Differential Geometry and then do General Relativity. Take this book as your first step to your final destination!

Overhyped
While this book is good, its a little overhyped. I did not particularly care for this book's presentation of connectedness and compactness (ie, the last two chapters), but the first three chapters were good. The problems in this book were also pretty good. They were at least interesting and difficult. However, there are no solutions, so it might not be the best book for self study. I personally think introduction to topology by gamelin and greene is better. These books should be used in conjunction with topology by munkres.

very mindful of the student
I highly recommend this book.The problems are excellent.They really hit home and force you to truly understand the content.They get to the crux of the issues (some problems specifically test to make sure you didn't misinterpret a definition for example) and they're also interesting.

The book is carefully written in a simple style.It's a bit hard to explain...For lack of a better explanation, an analogy would be to how Mac computers are simple to use but not lacking in function.One specific example that I can pinpoint is that the author avoids using symbols excessively.

It is not a "layman" book at all however.Some problems take a lot of thinking.Some of them take me a few hours of scribbling in my notebooks and some of them take a few days of mulling over on top of that. But I'm not a math student or math practitioner (only a hobby at this point) so mathematicians-to-be should have an easier time than I.

Great book for self study
I'd like very much this book.The book is very conceptual and also rigorous.It isself-consistent and this facilate his study. It progress step by step.If you need an Introduction, for self study this is a right book for starting. His writting style is very clear and the edition is also very good. The only defect I found is that there is no solutions for the excersices.

An amazing read!
Absolutely great reading. It starts by explaining set theory more thorougly than many other introductory books, while it does it in a rigorous manner that prepares you for the rest of the chapters. I'm just a mathematics hobbyist, and I still have no problem grasping the content, while you can really appreciate the mathematical rigour. Great read. Go for it. ... Read more

Product Description
"A General Topology Workbook" presents elementary general topology in an unconventional way. Highly influenced by the legendary ideas of R.L. Moore, the author has taught several generations of mathematics students with these materials, proving again the usefulness and stimulation of the Moore method. The first part of the book gives a quick review of the basic definitions of the subject, interspersed with a large number of exercises, some of which are described as theorems. The second part contains complete solutions and complete proofs, providing students the opportunity to explore the details of solution and proof in comparison with what they have devised themselves. The book could be used for a moderately paced, one semester, upper division general topology course devoted to this method. ... Read more

Customer Reviews (1)

A concentrated, intense way to learn general topology
This book brings back memories of a graduate course in general topology that I took as an undergraduate, which was taught via the "Moore method", after the late Robert Lee Moore, who invented it. Handouts were given to the class (there were only 3 of us), and each of us was expected to work out or prove every result in the handout, without consulting references or collaborating with other students. Theorems were to be proved, or counterexamples given, but we did not know a priori which item from the handout was actually true or false. Needless to say this took a lot of work, and all of us had to present our results on the blackboard for scrutiny by both classmates and instructor.

The Moore method has its defenders and detractors. It certainly encourages originality of thought and strict intellectual honesty. Students can find incredible reinforcment as they discover that they can indeed give original proofs of sometimes very difficult (and famous) results in general topology. The downside is that not as much material is covered as compared to a traditional course in general topology. Students who are hungry to get to the frontiers of research might become impatient because of this.

This book does not follow the strict methodologies that we followed in our class, but instead reveals to the reader which results are true and then encourages their proof. Readers are also lead through the construction of examples and counterexamples, allowing them to gain more of the intuition needed for a thorough understanding of general topology. It is also a good book to use for independent study, as the answers to the results are given in the book (and this actually is the major portion of its bulk). ... Read more

Product Description
Coverage is fairly standard for a general topology course. Intended primarily for an undergraduate course. DLC: Topology. ... Read more

Customer Reviews (2)

Good book, but overprized.
This book indeed covers the standar subject of topology, and is well written, self contained and elegant. For a topology class there is no much more for the teacher to do than to clearify the definitions, so the best advice is to get the book in order to be prepared for class.

The only problem with this book is the prize. I don't understand why publishing companies think that we are going to pay 120 bucks for a paperback edition of a book on a subject that is not even "state of the art - frontiers of the science" kind.

Be At One With The Topological Cosmos
I used this textbook to supplement Dr. Cain's notes in a topology class at Georgia Tech.I found it to be an excellent introduction to the subject.Theorems are proved with clarity, and the exercises inspire thought about the subject matter.This book was the first place I encountered a proof of the Tychonoff Product Theorem. The proof was presented very well, in my opinion.In addition, there is no shortage of examples in each section.I would heartily recommend this book to anyone with an interest in learning general topology. ... Read more

Product Description
Over 140 examples, preceded by a succinct exposition of general topology and basic terminology. Each example treated as a whole. Over 25 Venn diagrams and charts summarize properties of the examples, while discussions of general methods of construction and change give readers insight into constructing counterexamples. Extensive collection of problems and exercises, correlated with examples. Bibliography. 1978 edition.
... Read more

Customer Reviews (10)

Resource for deep knowledge of Point-Set Topology
This book is a must have for anyone who wants to contribute to remaining questions in point-set topology.Property inheritance relationship diagrams fill the book, quickly giving someone a good knowledge of all the classic point-set properties of spaces more thoroughly than is ever taught in grad-school these days. The only draw back is that the book and counterexamples deal strictly with point-set.How nice it would be to have a new edition (or volume two) of this type of book pertaining to algebraic topology.For example, what is an example of a non-contractible space with all zero homotopy groups?This question (and any algebraic top. questions) wont be answered in Steen's book.

Counterexamples in Topology
I have found this book to be confusing to use and therefore of little to no value. If I had seen in a bookstore and not Online I would not have purchased it. I also purchased Schaum's Outline of General Topology which is very good.

a veritable mine of information....
To paraphrase Chandrasekhar's review of Watson's Bessel functions text, this is "a veritable mine of information... indispensable to those who have occasion to use point-set topology." I don't think this book is intended to be a text (& I think the authors say so), in which case it would be terrible because it doesn't explain the concepts very much. It's mostly a catalogue of every kind of set you can come up with, every kind of topology you can put on it, and what properties it has such as what T_i axioms the space satisfies, whether it's compact, para compact, etc etc. Most of the time such things are proven, but be prepared to think hard sometimes about the proofs or fill in details. I'm the kind of student where I have trouble understanding things which are highly 'counter-intuitive' so I had trouble proving things, even when I knew definitions, when I did topology for the first time last term. Once I saw this book though I got used to all the weird things in topology (like the ordered square, R in the lower-limit topology, Sorgenfrey plane, etc etc). This book is incredibly useful as a reference.

Essential if you want to be good in point set topology
A distinct characteristic of point set topology is that it builds on counterexamples. If you thumb through any PST text, many theorems are in the form "If the space T is A,B,C, then the space is X,Y,Z". The point of point set topology (pun unintended) is too determine what A,B,C are, and to weaken the hypothesis. "Can we take condition B out? Maybe hypothesis C can be weaken considerably?" How can we answer these questions? You're right, by counterexamples. Students who want to master point set topology should know the various counterexamples, no matter how contrived or unnatural they seem. While textbooks usually present a counterexample to show why Theorem Three Point Five Oh will not work on a weaker assumption -- most students (and teachers) tend to skip these parts. A collection of counterexamples presented in this book (excellent organisation, by the way) is an essential supplement of a topology course; it enables one to 'see' between the points, so to speak.

a good book to combine with a regular textbook
This book has examples in it that are "missing", so to speak, from many regular topology books. It aims to shore up some of these shortcomings, with examples that the student can see and understand. There are charts and graphs, as well as a detailed explanation. Some "problems" often found in regular topology books are solved. Very few proofs, if any, are given. This is not a book meant to be studied without a regular textbook on topology, only to be used as an overall review of problems and short basic premises of topology. Use this in addition to your regular fare, but keep it close at hand when doing homework or preparing for an exam.
There are fundamentals on Cantor's Theorem, the countability or uncountability of sets, compactness, closed and bounded functions, open sets, continuity, connectedness, etc. All these are basic to topology, and this book does address them, but in a brief way. It then shows a basic overview of topology that helps greatly to understand the different fields of topology. ... Read more

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Translated and Revised by C. Cecilia Krieger. Critically acclaimed text presents detailed theory of Fréchet (V) spaces and a comprehensive examination of their relevance to topological spaces, plus in-depth discussions of metric and complete spaces. Numerous exercises reinforce teachings of each chapter. "...an elegant piece of work suitable for the beginning student and the mature mathematician."—Scripta Mathematica. 2nd ed.
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Customer Reviews (1)

An excellent reference book
I found Engelking a difficult book to study general topology from, but an excellent reference book. Engelking covers wide areas of General topology, starting from the basic definition. At times the exposition seemed to veer into too much technicalities, which seemed to distract one, if one was reading the book alone. What makes the book truly great, however, is its index, and bibliography. It's wealth of examples and counterexamples is very useful, as well.

Finally, the Russian translation seems to be enriched with the comments of translators and more bibliography. ... Read more

Product Description
This classic work has been fundamentally revised to take account of recent developments in general topology. The first three chapters remain unchanged except for numerous minor corrections and additional exercises, but chapters IV-VII and the new chapter VIII cover the rapid changes that have occurred since 1968 when the first edition appeared.

The reader will find many new topics in chapters IV-VIII, e.g. theory of Wallmann-Shanin's compactification, realcompact space, various generalizations of paracompactness, generalized metric spaces, Dugundji type extension theory, linearly ordered topological space, theory of cardinal functions, dyadic space, etc., that were, in the author's opinion, mostly special or isolated topics some twenty years ago but now settle down into the mainstream of general topology.

... Read more

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Intended for use both as a text and a reference, this book is an exposition of the fundamental ideas of algebraic topology. The first third of the book covers the fundamental group, its definition and its application in the study of covering spaces. The focus then turns to homology theory, including cohomology, cup products, cohomology operations, and topological manifolds. The remaining third of the book is devoted to Homotropy theory, covering basic facts about homotropy groups, applications to obstruction theory, and computations of homotropy groups of spheres. In the later parts, the main emphasis is on the application to geometry of the algebraic tools developed earlier. ... Read more

Customer Reviews (6)

Pioneering text
This book was an incredible step forward when it was written (1962-1963). Lefschetz's Algebraic Topology (Colloquium Pbns. Series, Vol 27) was the main text at the time. A large number of other good to great books on the subject have appeared since then, so a review for current readers needs to address two separate issues: its suitability as a textbook and its mathematical content.
I took the course from Mr. Spanier at Berkeley a decade after the text was written.He was a fantastic teacher - one of the two best I've ever had (the other taught nonlinear circuit theory). We did NOT use this text, except as a reference and problem source. He had pretty much abandonded the extreme abstract categorical approach by then.The notes I have follow the topical pattern of the book, but are so modified as to be essentially a different book, especially after covering spaces and the first homotopy group. His statement was that his treatment had changed since the subject had changed significantly.So much more has changed since then that I would not recommend this book as a primary text these days. Bredon's Topology and Geometry (Graduate Texts in Mathematics) is much better suited to today's student.
So, why did I give it four stars?First, notice that it splits stylewise into three segments, corresponding the treatment of its material in a three quarter academic year.The first three chapters (intro, covering spaces, polyhedral) have really not been superceded in a beginning text.Topics are covered very thoroughly, aiding the student new to the subject.The next three chapters (homology) are written much with much less explanation included - indeed, some areas leave much to the reader to discover and, consequently, aren't very helpful if the instructor doesn't fill in the details (the text expects a rather rapid mathematical maturation from the first part - too much of a ramp in my opinion), but the text is comprehensive.The last section (homotopy theory, obstruction theory and spectral sequences) should just be treated as a reference - it'd be hard to find all this material in such a compact form elsewhere and the obstruction theory section has fantastic coverage of what was known as of the writing of this book.It's way too terse for a novice to learn from and there are some great books out there these days on the material.

For reference ONLY
This book is a highly advanced and very formal treatment of algebraic topology and meant for researchers who already have considerable background in the subject. A category-theoretic functorial point of view is stressed throughout the book, and the author himself states that the title of the book could have been "Functorial Topology". It serves best as a reference book, although there are problem sets at the end of each chapter.

After a brief introduction to set theory, general topology, and algebra, homotopy and the fundamental group are covered in Chapter 1. Categories and functors are defined, and some examples are given, but the reader will have to consult the literature for an in-depth discussion. Homotopy is introduced as an equivalence class of maps between topological pairs. Fixing a base point allows the author to define H-spaces, but he does not motivate the real need for using pointed spaces, namely as a way of obtaining the composition law for the loops in the fundamental group. By suitable use of the reduced join, reduced product, and reduced suspension, the author shows how to obtain H-groups and H co-groups. The fundamental group is defined in the last section of the chapter, and the author does clarify the non-uniqueness of the fundamental group based at different points of a path-connected space.

Covering spaces and fibrations are discussed in the next chapter. The author does a fairly good job of discussing these, and does a very good job of motivating the definition of a fiber bundle as a generalized covering space where the "fiber" is not discrete. The fundamental group is used to classify covering spaces.

In chapter 3 the author gets down to the task of computing the fundamental group of a space using polyhedra. Although this subject is intensely geometrical. only six diagrams are included in the discussion.

Homology is introduced via a categorical approach in the next chapter. Singular homology on the category of topological pairs and simplicial homology on the category of simplicial pairs. The author begins the chapter with a nice intuitive discussion, but then quickly runs off to an extremely abstract definition-theorem-proof treatment of homology theory. The discussion reads like one straight out of a book on homological algebra.

This approach is even more apparent in the next chapter, where homology theory is extended to general coefficient groups. The Steenrod squaring operations, which have a beautiful geometric interpretation, are instead treated in this chapter as cohomology operations. The logic used is impeccable but the real understanding gained is severely lacking.

General cohomology theory is treated in the next chapter with the duality between homology and cohomology investigated via the slant product. Characteristic classes, so important in applications, are discussed using algebraic constructions via the cup product and Steenrod squares. Characteristic classes do have a nice geometric interpretation, but this is totally masked in the discussion here.

The higher homotopy groups and CW complexesare discussed in Chapter 7, but again, the functorial approach used here totally obscures the underlying geometrical constructions.

Obstruction theory is the subject of Chapter8, with Eilenberg-Maclane spaces leading off the discussion. The author does give some motivation in the first few paragraphs on how obstructions arise as an impediment to a lifting of a map, but an explicit, concrete example is what is needed here.

The last chapter covers spectral sequences as applied to homotopy groups of spheres. More homological algebra again, and the same material could be obtained (and in more detail) in a book on that subject.

Definitely not for beginners
I gave Spanier only three stars not because I think it is a bad book: as the previous two reviewers have pointed out, Spanier is a comprehensive (and still good) account of the subject, but is by no means for beginners. Now that more user-frinedly ones like Bredon, Fomenko-Novikov, and Hatcher (forthcoming) are available,it would hardly justify giving it four or five stars.And for reference purposes, there is a small (and sometimes too terse) but attractive account by May that covers topics not touched by Spanier.

Excellent reference, poor textbook
This book is terrific as a reference for those who already know the subject, but if you teach algebraic topology it would be dangerous to use it as a graduate text (unless you're willing to supplement it extensively).The basic problem is that Spanier does not teach students how to computeeffectively because his abstract, high-powered algebraic approach obscuresthe underlying geometry, which is not developed at all. Here I'd recommendthe books by Munkres, or Greenberg; even the old-fashioned treatment ofLefschetz, with its explicit and rather cumbersome treatment of cohomology,could serve as an antidote to Spanier. Somewhere, the student has toacquire a good intuitive feeling for the geometry underlying the subject(the same can be said of algebraic geometry -- here earlier work (e.g., ofthe Italian school, Weil's old book on intersection theory, ...) should notbe neglected entirely in favor of Grothendieck et al., for somethingessential is lost)

That said, if you already know the subject Spanier'sbook is an excellent reference. Even here, though, you'll need to providesome details toward the ends of the later chapters. Each chapter starts outrelatively easily and works up to a crescendo, the treatment becomingterser and more advanced.

I give it four stars (5 for mathematicalquality, 3 for usefulness as a text). The first three chapters deal withcovering spaces and fibrations; the middle three with (co)homology andduality; the last three with general homotopy theory, obstruction theory,and spectral sequences. Some of Serre's classical results on finitenesstheorems for homotopy groups are presented.

Excellent reference, poor textbook
This book is terrific as a reference for those who already know thesubject, but if you teach algebraic topology it would be dangerous to useit as a graduate text (unless you're willing to supplement it extensively). The basic problem is that Spanier does not teach students how to computeeffectively because his abstract, high-powered algebraic approach obscuresthe underlying geometry, which is not developed at all.Here I'd recommendthe books by Munkres, or Greenberg; even the old-fashioned treatment ofLefschetz, with its explicit and rather cumbersome treatment of cohomology,could serve as an antidote to Spanier.Somewhere, the student has toacquire a good intuitive feeling for the geometry underlying the subject(the same can be said of algebraic geometry -- here earlier work (e.g., ofthe Italian school, Weil's old book on intersection theory, ...) should notbe neglected entirely in favor of Grothendieck et al., for somethingessential is lost)

That said, if you already know the subject Spanier'sbook is an excellent reference.Even here, though, you'll need to providesome details toward the ends of the later chapters.Each chapter startsout relatively easily and works up to a crescendo, the treatment becomingterser and more advanced.

I give it four stars (5 for mathematicalquality, 3 for usefulness as a text).The first three chapters deal withcovering spaces and fibrations; the middle three with (co)homology andduality; the last three with general homotopy theory, obstruction theory,and spectral sequences.Some of Serre's classical results on finitenesstheorems for homotopy groups are presented. ... Read more

Customer Reviews (1)

Efficient.
This is an oustanding book on introductory topology (point set topology). The book is short and the author does not get into basic algebraic topology as Munkres, for example, does. However, it is solid and complete and the proofs presented by Dixmier are surprisingly optimised: very concise but always clear. Not only is this convenient while studying from it, but the style is also a model on how one should write one's proofs.

For those able to read mathematics in French, I'd strongly recommend the original version, if available. ... Read more

Product Description
This volume mainly focuses on various comprehensive topologicaltheories, with the exception of a paper on combinatorial topologyversus point-set topology by I.M. James and a paper on the history ofthe normal Moore space problem by P. Nyikos.The history of the following theories is given: pointfree topology,locale and frame theory (P. Johnstone), non-symmetric distances intopology (H.-P. Kunzi), categorical topology and topologicalconstructs (E. Lowen-Colebunders and B. Lowen), topological groups (M.G. Tkacenko) and finally shape theory (S. Marde J. Segal).Together with the first two volumes, this work focuses on the historyof topology, in all its aspects. It is unique and presents importantviews and insights into the problems and development of topologicaltheories and applications of topological concepts, and into the lifeand work of topologists. As such, it will encourage not only furtherstudy in the history of the subject, but also further mathematicalresearch in the field. It is an invaluable tool for topologyresearchers and topology teachers throughout the mathematical world. ... Read more

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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. ... Read more

Customer Reviews (32)

How good can a textbookbe?
Depth trough luminous exposition and a pletora of exceptional examples is still a main virtue of a Munkres (Basic?!) Topology opus. The expansion on Algebraic Topology was a definitely better crown for an already very well built textbook which does not seem to age.

Excellent
The book was delievered very quickly and was in excellent condition just like the seller suggested. I would definitely buy from this seller again.

Incredible textbook
I originally used this text in my undergraduate topology class. In the time since then, I have repeatedly returned to this book for reference. While I am specializing in algebraic number theory and algebraic geometry, I find that minor topological considerations still arise fairly often. I also used Munkres as my primary object of study for the topology qualifying exam.

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.

came in good condition
book was in great condition when i recieved it.i was worried that i would not get the book before my class started but that was my fault for ordering too late, it was shipped within the time frame that was posted.

Well written, but not interesting.
I read the first edition of this book quite seriously as an undergraduate. Although I found the presentation to be quite dry, at least it is clear throughout.

But now that I have made it through grad school, I have to object to Munkres' choice of material. (I am refering to the point-set portion of the book.) He covers all sorts of topics such as regular, normal, and Lindelof spaces, the Tietze extension theorem, etc. -- which I never saw again. Furthermore, he goes into all sorts of "pathological" counterexamples which in retrospect are really not interesting. It seems that Munkres' book would be quite useful as an introduction to set-theoretic topology and logic, and quite interesting from that point of view. But he doesn't seem to be gearing his book towards students who will study analysis, algebraic geometry, representation theory, algebraic number theory, differential geometry, etc. -- in other words 90% of math graduate students.

He makes his book self-contained, which in some sense is the problem -- in my opinion the interesting examples are the Zariski topology, infinite Galois theory, adeles and ideles, and the like. He doesn't address these, and perhaps these can't or shouldn't be covered in a book such as this. But in retrospect the time I spent studying this book just doesn't seem worthwhile.

Regrettably I have no particular alternative in mind to recommend. ... Read more

Product Description
This is a postgraduate level textbook, specifically written for use at universities in India and is designed to fill a long-felt need. Some of the welcome features of this book are: motivation and heuristic explanations which are provided before introducing some of the concepts; bases and subbases of a topology which are given more attention than in most books; both filters and nets are treated in great detail, along with a detailed discussion on the so-called equivalence of the two concepts; a detailed treatment of image and quotient topologies; compactification is also treated in far greater detail than in any other book, with ample remarks about common pitfalls; one-point compactifications other than Alexandroff are also considered; chapters on uniform spaces and function spaces proceed at a pace that the student will find easier to handle; a very large number and a broad spectrum of exercises, including the "drill work" type as well research-oriented type. The book also has a quick summary of basic set theory and a list of a large number of set-theoretic identities which the student should find very handy as a reference list. ... Read more

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Customer Reviews (2)

A Classic Work in Topology for The Undergraduate and Me, the Amateur!
Dear Readers

Author has taken great pains and made great efforts to "push" the student along with a plethora of examples and homework problems, while at the same time plowing forward on the trail to the great generalization of non-metrizability.

Those "nasty" collections of open sets each with their nasty collection of "interior points". A concept( interior point ) that took me forty years to understand. Once you understand the concept of open set and interior point the whole thing begins to "fall into place".

Then there's that nasty concept of "point of accumulation".( if its tough see Courant vol. 1 Courant is really a master at demonstarating this). Again, all is explained in this book by Kasriel with ample examples and homework questions abounding.

Please note: "geek" is not a member of the "set".

More later.

With Best Regards

Southern Jameson West

a great complement to any intro analysis or topology course
Having taken all of real analysis, complex analysis, and topology in arow, I have found that using outside references is key in a completeunderstanding of the given subject.This book has been my savior througheach of those classes.His explanations and proofs are extremely helpfuland he touches on every useful aspect of point set topology.The book israther thin, and I would have hoped for additional sections on the producttopology, but all the material he does cover, he covers well.This is abook you should definitely add to your mathematics collection. ... Read more