Book Description

Customer Reviews (7)

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.

Topology encyclopedia
Every mathematician need this book if he want to learn topology. Is a classic. You can use it for learn and as a reference. Further, this book has a extensive bibliography.

Wilard's Topology
After looking at several books on topology, I would have to say that Wilard's General Topology is an excellent resource book.For those who have taken a topology course and want a little more practice with problems, this book has numerious exercises that help form an solid knowledge base.What else is nice about the problems is they are good research-starters for undergraduates.The examples in the chapters are non-trivial and explain the ideas of the chapter.Also, Wilard's General Topology has a slight set-theoritic view to topology, so those who like set-theory and topology, this book will be of great use.I suggest Wilard's General Topology if you need another topology book to help explain ideas from class or other books.

Very terse
I found this book very terse.It seems like it could be a great book for someone already familar with topology, I don't recremend it for anyone who is not very familar with the subject.It lacks examples of how to solve problems, then throws quite complicated ones at you very quickly.A few mistakes in printing also. ... Read more

Book 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 (4)

Very good
The book is wonderful, recommends to all. Very good

Geraldo Tavares -Campinas/SP- Brasil.

OK as a supliment
If you're taking a basic topology class (or any other abstract math class), it is usefull to have other books to reference.This book was only OK as a supliment.I found that you had to be careful the definitions, as some topologists will axiomize ideas differently (I found this happened the most when dealing with seperation axioms).I found that there are a few dover books out there that are a little more readable (Gemignani's, Mendelson's and Willard's books all come to my mind).This book won't hurt, but there are other options within it's price range.

Great resource for college-level math students...
I wish I had known about this summary when I was taking my Fundamental Concepts in Math course at the university.I would have done a lot better!I agree with the previous reviewer about adding REA's Problem Solver book for a complete set.

The topics covered include: (1) Sets and relations, (2) Functions, (3) Cardinality and Order, (4) Topology of the line and plane, (5) Topological Spaces, (6) Bases and Subbases [sic], (7) Continuity and topological equivalence, (7) Metric and normed spaces, (8)Countability, (10) Separation axioms, (11) Compactness, (12) Product spaces, (13) Connectedness, (14) Complete metric spaces, and (15) Function spaces.There is an appendix about Properties of real numbers.

This is a technical book and not for a casual reader.I believe this will be difficult for a teach-yourself-to-know reader, unless the reader has considerable math experience.

Another book that I found helpful in Fundamental Concepts (not simply Topology) was Courant et al., What is mathematics? 2nd ed.

My Fundamental Concepts professor taught the course with only lecture notes, and I was completely lost.Had I had this outline, I think I would have done better, because this Outline has a number of solved problems and exercises.

Good luck.

If you are a good researcher, this is a good resource!
If you are taking an introductory Topology course, I recommend using this book. It is a little old, and some of the problems are not solved, but it will offer you an advantage when using your class text. You should use Lipschutz's book together with REA's Topology Problem Solver to give you full study advantage when taking on this most difficult subject. With some moderate study, you should do better on your assignements! ... Read more

Book Description
This classic book is a systematic exposition of general topology. It is especially intended as background for modern analysis. Based on lectures given at the University of Chicago, the University of California and Tulane University, this book is intended to be a reference and a text. As a reference work, it offers a reasonably complete coverage of the area, and this has resulted in a more extended treatment than would normally be given in a course. As a text, however, the exposition in the eariler chapters proceeds at a more pedestrian pace. A preliminary chapter covers those topics requisite to the main body of work. ... 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

Book 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

Book Description

This introduction to topology provides separate,in-depth coverage of both general topology and algebraic topology.Includes many examples and figures. GENERALTOPOLOGY. Set Theory and Logic. Topological Spaces and ContinuousFunctions. Connectedness and Compactness.Countability and SeparationAxioms. The Tychonoff Theorem. Metrization Theorems andparacompactness. Complete Metric Spaces and Function Spaces. BaireSpaces and Dimension Theory. ALGEBRAIC TOPOLOGY. The FundamentalGroup. Separation Theorems. The Seifert-van Kampen Theorem.Classification of Surfaces. Classification of Covering Spaces.Applications to Group Theory. For anyone needing abasic, thorough, introduction to general and algebraic topology andits applications.

Customer Reviews (34)

,,,
This is the standard into to point-set topology for a reason.It covers general topology very well, with easy-to-follow proofs and exercises that are actually possible to do.The problem is that the algebraic topology portion of this text (around 1/3 of the whole thing), is vastly inferior to Hatcher's "Algebraic Topology" book, which happens to be free.If you're looking for a good way to begin studying topology, then this is the book you're looking for, but if you want to learn about the fundamental group (and various related topics), then Allen Hatcher's book is available for free on his website.It calls to you...[insert spooky noises here]

Ideal
I've gone through most of this book and did many of the problems. The sections I skipped are the section on nets, the "review section" in chapter 4, the existence of a continuous nowhere-differentiable function, Dimension Theory, and all of Chapter 10 (Separation Theorems in the Plane), the Classification Theorem, and Constructing Compact Surfaces.

This is definitely my favorite math book. The two other books I've read this semester (Conway's Complex Analysis, and Rudin's Real and Complex Analysis) simply don't compare. In fact I'm afraid I'll always find fault with every other math book, after reading this one. There's alot of good expository prose, many examples and diagrams, and if you pay attention to details, and struggle to supply missing ones, you won't miss a beat and will succeed (unlike sometimes in Rudin's text). The problems are appropriate; very few are mindless, most do require a little thought, but a motivated student could solve most or all of them in a reasonable amount of time. There are no sudden breaks in proofs or in the text that are relegated as exercises (unless it's a repeat of a previous proof), and although results from previous exercises are sometimes used, he always states the necessary hypotheses. The book is self-contained - he begins with 70+ pages of naive set theory, for instance (not a prerequisite for the rest of the book).

I feel that reading this book and working its problems has given me a solid and comprehensive grounding in basic topology, and this book does go beyond what's usually taught in a first topology course, and the second half of the book is all algebraic topology. Here I found the review of abelian groups, free products and free groups to be extremely helpful, though I did still have to contemplate these alot on my own afterwards. The Seifert-Van Kampen theorem was also well-presented; he presents it as a pushout diagram. In the last chapter, as a nice application, he proves using linear graphs that subgroups of free groups are free.

I just simply love this book, but to be fair, I do have some minor qualms.

(1) There are a few obvious typos, and I didn't find more than six
(2) I believe one step in the proof of Lemma 68.9 is incorrect; this arises from a definitional issue of the subgroup generated by a subset. earlier, he assumed the subset was itself a subgroup, but now he's assuming it's arbitrary. the correct definition is on the next page, and the method of proof, with this definition, does give the right result; almost nothing changes in the proof
(3) In Theorem 68.4, the monomorphism and generating assumptions aren't necessary
(4) Problem #2 on page 438: I think the X_i should be path-connected, and Wikipedia is in agreement with this. I tried passing to path-components, which solved one problem but gave me others. On the other hand, if you assume path-connectedness, the proof is is the right level of difficulty.
(5) He gives an exercise regarding absolute retracts and adjunction spaces. I think he should've elaborated more on adjunction spaces, as it does involve new notions (e.g. free/topological union). Also his definition of adjunction space is incomplete, as compared to other definitions I found
(6) The book binding is horrible (it's the same with his other book, "Analysis on Manifolds"). If you're paying 100+ dollars for a book, you should expect to receive something very pretty, but the typesetting of this book is quite dull, and the book falls apart easily (mine is in many pieces).

In conclusion I highly recommend this book for self-study, and for seeing how math books can and should be written. I hope Munkres writes more textbooks, I'd read every single one of them.

Good for auto-didacts
I used this book to teach myself some topology. Not being a mathematician, I cannot really assess how it trades off rigour with accessibility, but I can recommend it for self-study. It starts more or less from zero, is pretty clear and provides some welcome intuition to supplement the proofs. The best thing about it is the large number of challenging exercises, solutions to which are readily available on the web ([...]).

First half is great, don't bother with the second half
There are 14 chapters in the book, but it is only known for the first 8.The first 8 chapters cover pretty much everything you ever wanted to, or will ever need to know for point-set topology.It's easy to read, makes sense, lots of examples, proofs, and doable exercises.Extremely thorough.Munkres likes to talk, and some of his informal language is pretty funny in some places.Go somewhere else for Algebraic Topology.Some basics on homotopy theory are here, but nothing at all on (co)homology.My main complaint about this book is that it fails to make topology as exciting as it really is.

The best approach to point-set topology and an excellent graduate text
This is my *absolute* favourite maths text of all time. In the years I've owned it, I have recommended this book to nearly every single individual in mathematics I know. The approach of the book is unlike other books on point-set topology. Most books begin with metric spaces, and build up the motivation towards general topology. However, this book drops you right into the deep end, and begins with the general topology. The main philosophy seems to be not about "metric" but about "metrisable." Munkres covers everything you would possibly want from a book on general topology. In part 2, he even delves a little into Algebraic topology. It has been years since I first opened my copy of Munkres, and I keep referring again and again! ... Read more

Book Description

If mathematics is a language, then taking a topology course at the undergraduate level is cramming vocabulary and memorizing irregular verbs: a necessary, but not always exciting exercise one has to go through before one can read great works of literature in the original language.

Customer Reviews (1)

Flawless exposition, great examples, short enough to read cover to cover
This skinny little math book from the Springer Universitext series achieves excellence on many levels.First of all, anyone familiar with the old quip "A topologist is someone who cannot tell the difference between a coffee mug and a donut" will instantly smile when they see the cover.The exposition is downright beautiful, and the organization of the material could not be more perfect.The remarkable thing is that the examples not only demonstrate the concepts, but also play a large role in the development.The choice of fonts and notation is well thought-out and, although minor, contributes greatly to the excellence of the book.One of the best features of this book is its length.With less than 200 pages, one can reasonably set a goal to read it cover to cover.The well-chosen examples not only aid in understanding, but also serve to introduce the reader to concepts from other areas of mathematics.On that note, not only those seeking an introduction to topology, but also anyone new to advanced mathematics, and in addition seasoned mathematicians who are thinking about writing books themselves, will benefit greatly from reading this book.

The author divides the material into five chapters-- 1. Set Theory, 2. Metric Spaces, 3. Topological Spaces, 4. Function Spaces, and 5. Basic Algebraic Topology.There are a number of good exmples from chapters 2 and 3 that serve to compare and contrast properties of metric spaces and topological spaces, as can be expected in any topology text, however the examples used here are interesting in their own right in other areas of math.The author uses the Zariski topology on the prime ideals of a commutative ring in many places.The reader will also meet various function spaces and see how pointwise vs. uniform convergence manifest themselves through suitably chosen topologies.

A number of unique features worth noting are the proof of the Baire category theorem, which is derived from the so called Mittag-Leffler theorem (this is probably the only introductory text which proves this), and Tychonoff's theorem is proved using nets by expressing compactness as every net has a convergent subnet.Also of interest are proofs of the Stone-Weierstrass theorem and the Arzela-Ascoli theorem.On top of all this, there is still some room left at the end to introduce some basic homotopy theory.The fundamental group is defined and covering spaces are introduced.The author proves that homotopy-equivalent spaces have isomorphic fundamental groups, shows that paths and path homotopies can be lifted, and uses this to establish that the fundamental group of the circle is isomorphic to the integers.This is used to prove the Brouwer fixed-point theorem. ... Read more

Book Description
This is the softcover reprint of the English translation of 1971 (available from Springer since 1989) of the first 4 chapters of Bourbaki's Topologie générale. It gives all the basics of the subject, starting from definitions. Important classes of topological spaces are studied, uniform structures are introduced and applied to topological groups. Real numbers are constructed and their properties established. Part II, comprising the later chapters, Ch. 5-10, is also available in English in softcover. ... Read more

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.

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

Book Description

Taking a direct route, Essential Topology brings the most important aspects of modern topology within reach of a second-year undergraduate student. Based on courses given at the University of Wales Swansea, it begins with a discussion of continuity and, by way of many examples, leads to the celebrated "Hairy Ball theorem" and on to homotopy and homology: the cornerstones of contemporary algebraic topology.

Customer Reviews (2)

Best undergraduate topology book
I have never seen such a beatiful explanation on continuity and its relations to series and sets. Now I understand why, when mathematics is lousily explained,everything seemms to be so hard. I recommend strongly this book for someone for self study on topology. Hope the author can write on other topics of mathematics.

A pleasure to read
I have a major in math, many years ago. I have moved into economics, but miss the elegance of math, hence I decided to revisit some old topics, and started with topology. As a student we used lecture notes and no real textbook, so my choice now was this textbook. It is a pure pleasure to read. I wish we had used it as a text book when I studied.

The topics are well motivated. Crossley does a good job in explaining why we should care about these particular lemmas and theorems. The proofs are usually elegant. I find the estetic pleasures a good math book should provide. ... Read more

Book Description
This book constitutes nothing less than an up-to-date survey of the whole field of topology (with the exception of "general (set-theoretic) topology"), or, in the words of Novikov himself, of what was termed at the end of the 19th century "Analysis Situs", and subsequently diversified into the various subfields of combinatorial, algebraic, differential, homotopic, and geometric topology. It gives an overview of these subfields, beginning with the elements and proceeding right up to the present frontiers of research. Thus one finds here the whole range of topological concepts from fibre spaces, CW-complexes, homology and homotopy, through bordism theory and K-theory to the Adams-Novikov spectral sequence, and an exhaustive (but necessarily concentrated) survey of the theory of manifolds. An appendix sketching the recent impressive developments in the theory of knots and links and low-dimensional topology generally, brings the survey right up to the present. This work is the flagship of the topology subseries of the Encyclopaedia. ... Read more

Book Description

Book Description
This is the softcover reprint of the English translation of 1974 (available from Springer since 1989) of the later chapters of Bourbaki's Topologie générale. It completes the treatment of general topology begun in Part I (Ch. 1-4, also available in English in softcover). The real numbers having been introduced in Ch. 4, the first chapters of this volume study subgroups and quotients of R (with applications to the 'measurement of magnitudes' and to the log and exp functions), then real vector spaces and projective spaces, then the additive groups Rn (subgroups, quotients, homomorphisms, infinite sums and products). Analogous properties are then studied for complex numbers, in Ch.8. Chapter 9 illustrates the use of real numbers in general topology, studying different important kinds of topological spaces: uniformizable, metric, normal Baire, Polish, Borel spaces.The final chapter deals with the various topologies of function spaces,ending with a section on approximation of functions. ... Read more

Book Description

Customer Reviews (7)

Decent book with flaws
The book has its virtues, sure enough. But there are some downsides
to it as well that I feel are underrepresented in the other reviews so far.

Let me first note that, contrary to the statement of one other reviewer, there are exercises in this book, and not too few. However, I found that I did not need them, since thinking deeply about all the little flaws and omissions that are scattered through the text allowed me to mature faster than going through these exercises. Needless to say, though, that this type of exercise can be a bit frustrating. I often found myself wondering if it was my lack of maturity that made me struggle, or if the authors actually made their life too simple at various points. Luckily, I found amply evidence for the latter. For example, the reader familiar with homotopy may open the book on page 164 and inspect their proof that the curve given by f(1-x) is the inverse of that given by f(x) in the fundamental group. While this is a true statement of course, their constructed homotopy to prove this is not really continuous, and a slight modification of it could be used as a "proof" that every curve is homotopy equivalent to a constant one. A useful review of the book by a professional can be found at the following URL,

http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183524657

where similar shortcomings are noted. I agree that the latter will probably not slow down an expert who chooses this book as a reference. For beginners, however, they are unnecessary obstacles. I bought this book because I got attracted by the balanced selection of topics ranging from point set topology to algebraic topology. I wanted to learn the latter, but first needed to become proficient in the former. Having now read only the first part of the book devoted to point set topology, I can say that the book did its job, and did it quite well. However, I cannot shake off the feeling that I could have learned the same material in a fraction of the time from a different book. Feeling that I do now have a solid enough background in point set topology, I am considering to not read the second half of the book, and instead learn algebraic topology from a more modern text.

Very Impressed
I am teaching myself topology with this book right now, and I must say it has an excellent balance of motivation and rigor. The very first definition in the book reveals the implications of topology to anyone who has studied limit pts (and how connectedness is defined in terms of same). After less than a week of study, I understood the big picture better than most people I know who have taken a full course. The exercises are a little sparse, perhaps, but they generally make up for their small number with increased difficulty. I have only encountered a few exercises that I could call trivial. My only gripe is that the exercises are sometimes a little tricky to find.

A good start
Very clearly written, full of examples and counterexamples, making use of pictures but never sacrificing rigor, the authors of this book have given students of topology a superb introduction to the field. Many students have been educated in topology by using this book, and it is sure to remain a classic in the field. It builds a solid understanding of the basic rudiments and intuition behind point-set, geometric, and algebraic topology. There is a lot of material covered in the book, and some very specialized subjects, such as Cech and Vietoris homology and some dimension theory, but with some preserverance and concentration, the entire book can be grasped within reasonable time constraints. Probably the only minus to the book is the lack of exercises. This is a quite serious omission, for the only way to master a subject is to work problems that require careful thought for their solution.

The beginning student of topology should probably read this book with the following mindset: try to think of ways and techniques that you would devise to study the structure of a topological space. Homotopy and homology (in various forms) are the standard techniques for doing this. These strategies have varying degrees of success, but their use in topology now seems to be reaching a saturation limit, even though the explicit calculation of homotopy groups is still a very active area. New techniques and concepts, representing sort of a "large deviation" from the standard ones discussed in this book, will be needed to make further progress in the study of complicated topological spaces. Something more is needed now, that is completely different than homology and homotopy theory, that will make more transparent the properties of these spaces. These new techniques will be somewhat radical from the standpoint of current ones, but they will be more effective from a conceptual (and computational) point of view.

A Professional Topologist loves this book.
When I was a graduate student 40 years ago there were very few texts in topology.The only two that I recall being in use were Hocking and Young and the book by Kelley.Over the years my copy of Hocking and Young has become quite worn.It is a wonderful book that gives the true flavor of topology.It is also contains a large number of topics that one can refer to later on.It becomes quite apparent very earlier that no one will be able to fully appreciate the book in the time span of one course.It is a book that must be read and reread over and over again.It is a real classic.I do not believe that it is the type of book that would be of much or any general interest but to a point set topologist it is a classic and must for his bookself.I am quite surprised over its low price.I can not help but compare it with the newer book by Munkres.I recall seeing Munkres book many years ago and disliking it.But the current edition seems much closer in flavor to HY and Munkres book is quite good.Munkres style is much clearer than HY, but both books target a very specialized group of people.Neither book is for the faint of heart and will take many years to absorb.Considering that Munkres book is 9 times as expensive as HY, HY seems to be the better buy.

Theoretical Dictionary
An excellent book, not for those persons unfamiliar with the topic of topolgy; yet, combined with simpler texts this book is a goldmine of topological theorems and their proofs. ... Read more

Book Description
This book is intended as a textbook for a first-year graduate course on algebraic topology, with as strong flavoring in smooth manifold theory. Starting with general topology, it discusses differentiable manifolds, cohomology, products and duality, the fundamental group, homology theory, and homotopy theory. It covers most of the topics all topologists will want students to see, including surfaces, Lie groups and fibre bundle theory. With a thoroughly modern point of view, it is the first truly new textbook in topology since Spanier, almost 25 years ago. Although the book is comprehensive, there is no attempt made to present the material in excessive generality, except where generality improves the efficiency and clarity of the presentation. ... Read more

Customer Reviews (6)

a different perspective
While I agree with reviewers generally that this is a good book, i should warn that bredon isnt for the faint of heart. He makes use of simple language from category theory, doesnt always completely introduce his discussions (see for example the chapter on the tangent bundle where tangent bundle is never defined), and some other things that are nuisances to the newcomer.

I do think this is a good modern readable textbook, but for the student who has a solid foundation in mathematics. I didnt find it as accessable as other topology books, say Hatcher or Lee's books (but lee's are not as complete).

excellent for first year graduate study
This was the assigned book in my first year grad topology course. It has good examples, interesting exercises. I like the emphasis on geometrical examples, constructions. It's not easy to read, but interesting.

Among the best textbooks in algebraic topology.
As the previous reviewers have commented, this book is very accessible and detailed. I should add that the authour never lets you get lost in the labyrinth of abstract nonsense; the treatment is always geometric rather than homologico-algebraic. The only complaint I have is, the book would be more useful with chapters on spectral sequences, cobordism and K-theory.

The Graduate Sudent's Topology Bible
If you want to learn topology, this book is the place.Though this text can require some maturity, the range of topics and the clarity of exposition are outstanding.My only complaint is that an additional appendix covering the basics of category theory would have been quite useful.Bredon not infrequently uses the language of category theory (though always in a non-essential way).Since this text is aimed at 1st year graduate students, I think the tacit assumption that the student has already encountered these topics is not justified.That such a minor point is my chief complaint speaks volumes of my esteem for this text.

My secret weapon in topology
Today I told someone that Bredon's "Topology and Geometry" book was my secret weapon.I say this because it has mostly everything a grad student in topology needs to know in order to be fluent in the subject.Ibought it not because I was taking a class from it--I got it because I sawit at the library and realized that this is the 'end all' topology book forme and that I will not need to buy all those 'other' books in topologyanymore.I will warn you though--some of the proofs are terse, so you haveto be somewhat 'mathematically mature'.It is definately harder thanMunkres' "Topology: A first course" (the red book).But it goesbeyond the point set crap that gets (in my opinion) too much airtime.Iwish I would have taken a class under this book. ... Read more

Book Description

Customer Reviews (6)

Intro to Topology
This is a wonderful book to start your topological studies with. It has manyproblems for one to do so one can practice and study and have the ability to make the grade on one's tests

Great book on Topology
I bought this book for my own enlightenment after already having a course in Topology here at Penn State University.What I find most interesting about this book is that the author explains the philosophy on the ideas and what we are really trying to say with these definitions and theorems.The book I used in my course didn't explain much at all so it would have been much more difficult to teach yourself from this book.Topology is somewhat abstract so if you're looking to study Topology this is a great book to start.A word of advice, read over a theorem and proof and try to reproduce it on paper from your mind.Help yourself from the book a bit along the way if necessary.You will learn much more this way as opposed to following along the proofs in the book as you read.You might also be interested in Counterexamples in Topology, a book with thousands of counterexamples.

Excellent Text for the Price
I chose this text for an independent study course in topology, but I ended up switching to Munkres fairly quickly. Considering the price it is an excellent text. All the fundamental topics of point-set topology are covered in a clear and orderly manner, but Mendelson treats metric spaces in much detail before he ever gets to definition of the topology. Some people may prefer this approach, but I think the definition should be introduced first.

Good Introduction to Metric Spaces and Topology
I was not a mathematics major, and only in recent years have I ventured into abstract mathematics. I was motivated to learn about topology as an aid to understanding a particular 3-D earth modeling application.

I read Introduction to Topology in three stages: as a review of set theory and metric spaces (chapters 1 and 2), then as an introduction to topology (chapter 3), and lastly as a detailed look at two important topological properties, connectedness (chapter 4) and compactness (chapter 5). I had previously read (and reviewed) another book titled Metric Spaces by Victor Bryant, but Mendelson is my first serious look at topology.

My reading of Mendelson - a 200-page text - required about 100 hours, substantially longer than the 40 to 60 hours estimated by an earlier reviewer. No solutions are provided for the section problems, which are generally proofs, not explicit problems.

The first chapter provides a concise overview of set theory and functions that is essential for Mendelson's later chapters on subsequent set-theoretic analysis of metric spaces and topology.

The second chapter is a solid introduction to metric spaces with good discussions on continuity, open balls and neighborhoods, limits from a metric space perspective, open sets and closed sets, subspaces, and equivalence of metric spaces. Chapter 2 concludes with a brief introduction to Hilbert space.

The third chapter introduces topological spaces as a generalization of metric spaces, and many theorems are largely restatements of the metric space theorems derived in chapter 2. I was thankful for this approach.

Mendelson begins chapter 3 by demonstrating that 1) open sets and neighborhoods are preserved in passing from a metric space to its associated topological space and 2) the existence of a one to one correspondence between the collection of all topological spaces and the collection of all neighborhood spaces.

He then reminds us that in a metric space we can say that there are points of a subset A arbitrarily close to a point x if the metric d(x, A) = 0. In characterizing this notion of arbitrary closeness in a topological space, Mendelson introduces the closure of A, the interior of A, and the boundary of A. Other topics included topological functions, continuity, homeomorphism (the equivalence relation), subspaces, and relative topology.The final sections in chapter 3 on products of topological spaces, identification topologies, and categories and functors were more difficult.

In chapter 4 the initial sections (connectedness on the real line, the intermediate value theorem, and fixed point theorems) were largely familiar. But thereafter I became bogged down with the discussions of path-connected topological spaces, especially with the longer proofs involving the concepts of homotopic paths, the fundamental group, and simple connectedness.

Chapter 5, titled Compactness, was even more abstract and difficult, with topics like coverings, finite coverings, subcoverings, compactness, compactness on the real line, products of compact spaces, compact metric spaces, the Lebesgue number, the Bolzano-Weierstrass property, and countability. Perhaps, a reader more familiar with analysis would have less difficulty with the last two chapters.

In summary, Introduction to Topology is quite useful for self-study. Mendelson's short text was intended for a one-semester undergraduate course, and it is thereby ideal for readers that either require a basic introduction to topology, or need a quick review of material previously studied. The last two chapters on connectedness and compactness are substantially more difficult, but are still accessible to the persistent reader.

Ideal for self-study
This book is ideal for self-study. If you have not had the luxury of taking a topology course during your undergraduate studies, but you need to know some topology and you have to study it by yourself, this is the book you need. It is very readable and it explains carefully every concept. However, it is just an introductory text and it contains only basic material. You don't have to invest a lot of time to study the material in this book: let's say 40-60 hours of study are enough to grasp everything. I reccomend it especially to those graduate students of applied mathematics, finance, statistics or economics, who need to use some basic result from topology in their work. ... Read more

Book Description

Customer Reviews (6)

I'm not good at math
I wanted to teach myself some topology and a friend with mutiple Math PhD's reccomended this book to me.
This is a tremendous value, and is comprehensible. But it is prety lean and direct, so be prepared to work on this in a quiet place where you can concentrate for a sustained period of time. Proofs are direct, and expect you to be familiar with notation through all of Algebra.
I re-emphasize: there is zero, no, nada, blank, null coddeling here. Every single word, every single notation is important, and if you haven't read, marked, and inwardly digested each one it is a promise you will be lost in a page or two and have to go back. There is no fat here at all and the authors don't babysit you or expalin anything five different ways. This is direct on the coal face math.
Still, I knew only basic basic basic totpology before this, and now I have a vague understanding of all the major areas of further inquiry.
A very good value.

Okay, not great. Overall, I give it a C+
The exposition, while clear and not without attention to subtleties of the theory, is a little scattered. The metric spaces chapter is very good, but after that, it goes downhill. In particular, I was pretty disappointed that the mean value theorem was not proved as an application of connectedness. Everybody sees the mean value theorem in calculus, and the proof is really quite elegant. Also, a lot of important notions in topology are relegated to the exercises, and the rest of the exercises are like applications to analysis. It's kinda nice to be challenged to see the definitions in multiple ways through the exercises, but it would be nicer to get all the perspectives in the exposition, and be given exercises that would deepen one's understanding of the material.

All in all, this book feels like "topology as a branch of analysis" and only helps the reader to develop a modest working topological intuition. For readers interested in topology as its own subject, Munkres' book is the only book. For those readers desiring a more introductory approach, I found Mendelson's book to be an excellent introduction - the chapters on connectedness and compactness are thorough and quite helpful - though that book is lacking in that it doesn't discuss separation axioms at all, and contains few exercises. But that book is unique in that it despite its brevity it touches on metric spaces, categories, and the fundamental group.

If you're going to read this book, get a copy of Mendelson's book - it will flesh out your understanding of topology.

excellent introduction to topology
I used this book to teach myself the basics of point-set topology and homotopy theory. What makes this book so great is that the author doesn't waste words in delving into the heart of a concept, while providing insight into it. A good collection of interesting problems, most with solutions in the back of the book. This makes this book very good for self study. If you liked Rudin, you'll probably like this book as well, as it is written in a similar style. If someone knows of a better introduction, do let me know.

exceptionally well organized
This is a lean fast introduction to topology at the third or fourth year level. Pure math types only. The book is terse but the topics are selected with care and one things leads to the next. The proofs are sufficiently detailed. Nearly every exercise has a solution in the back. The clearest exposition of the fundamental group I've seen.

Good grad school prep.
This is the usual text for introductory Topology at UCLA, where I took the course.The authors (who teach at UCLA) have "if you haven't chewed through every syllable you are not learning" mentality.In short, the book is terse and demands a lot from the reader.Looking back, this was great preparation for graduate school and is probably the best philosophy for the serious undergrad.The book contains all of the information one needs for an introductory course, but absolutely no more.Not a single character is wasted on "extraneous" explanation.Be ready for battle when opening this one, but it's worth it. ... Read more

Book Description
In this broad introduction to topology, the author searches for topological invariants of spaces, together with techniques for calculating them.Students with knowledge of real analysis, elementary group theory, and linear algebra will quickly become familiar with a wide variety of techniques and applications involving point-set, geometric, and algebraic topology.Over 139 illustrations and more than 350 problems of various difficulties will help students gain a rounded understanding of the subject. ... Read more

Customer Reviews (10)

Bad, Bad Book
This book is terrible. The author doesn't denote important material at all! Sometimes the most important part of a section is contained in one poorly written sentence. This book is subpar. Both Munkres and Hatcher provide everything this book does, in fact much more so, and presents the material in much more rigor. I haven't seen a worse introductory book on the subject, though for some reason people who already know the material seem fond of the book.

I'd give it -5 stars if I could.

A very welcome, intuitive approach to topology
Many of the standard introductions to Topology (Munkres comes to mind) focus more on the logical flow of the material, and less on the motivation for the material.This book focuses on the motivation, but after the first few chapters, the logical development is sound too.

The Armstrong book starts out with some fairly advanced concepts, outlining some interesting topological results before giving the modern definition of topological spaces in terms of open sets.Typically, authors give the open set definition of a Topology at the outset, before explaining what topology really is, and without explaining why that definition is used or how it was developed.Armstrong instead shows the historical motivation of the subject, and actually leads the reader through this development, starting with the less elegant but more intuitive definition of spaces in terms of neighborhoods.The equivalent open set definition is then taken in chapter two.However, once things get going, this book does not move slowly at all--quotient spaces and the fundamental group are presented early and covered in depth, and it is not long before the reader encounters genuinely advanced material, in rigorous form.

It's true that this book doesn't cover the same amount of raw material that a book like the Munkres does, and it's true that the book does not follow the most concise logical order, but it offers history, motivation, and initial exposure to more interesting results.Perhaps more importantly, it develops the reader's intuition.In many ways, this book is a complement to the Munkres, and an enthusiastic self-learner would benefit greatly from using both books simultaneously.

At the same time, this book does get into some more advanced topics.It has a particularly clear exposition of simplicial homology.My last word of praise about this book is that although it gives lots of motivation, it is still very concise.I think it's hard to go wrong with this book.

Your Average topology student will be frustrated...
This text is very very difficult to read for people like me, your average topology student. A difficult subject to grasp, the layout of this book simply does not help organize the material. I have purchased several other books, that while they don't make topology easy, at least make it digestable. Pass on this book and go with Munkres.

An acceptable text
I would recommend reading with a highlighter and marking up a lot of the text because many definitions, points of interest, etc... are not set apart from regular text and it can be difficult locating the information you want/need to know on a particular page because of this. I have already highlighted a good deal of the book so that I can flip through the pages quickly and locate what I need.

There are plenty of exercises in the book of easy to medium difficulty, but certainly not many that I would call "hard."

The text is easy to read even if it is not organized as well as Munkres book. I don't think this is a book anyone would regret getting for learning topology for the first time, but as the title clearly indicates, this is not a book for people taking a second course in topology.

NO NO
The fact that the author does not explicitly define things is a bad enough reason to stay away from this book. If you only want a light treatment of point-set topology, go for Munkres, otherwise, Hatcher. ... Read more