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.
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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
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.
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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

Product Description
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
A fresh approach to introductory topology, this volume explains nontrivial applications of metric space topology to analysis, clearly establishing their relationship. Also, topics from elementary algebraic topology focus on concrete results with minimal algebraic formalism. The first two chapters consider metric space and point-set topology; the second two, algebraic topological material. 1983 ed. Solutions to Selected Exercises. List of Notations. Index. 51 illus.
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Customer Reviews (8)

Great book for starters
Introduction to Topology by Gamelin is a great book for starters. There are a considerable number of exercises with answer sugestions and most part of the material in the book is self contain. Thus, Introduction to Topology is a reference for those who are familiarising with this subject.

A Wonderful Book
I highly recommend this book as a supplement to another topology text since it has a lot of hints and answers. Students may not learn very from it because they may just copy the answers in the back of the text. It is a great book though.

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. ... 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
In most major universities one of the three or four basic first-year graduate mathematics courses is algebraic topology. This introductory text is suitable for use in a course on the subject or for self-study, featuring broad coverage and a readable exposition, with many examples and exercises.The four main chapters present the basics: fundamental group and covering spaces, homology and cohomology, higher homotopy groups, and homotopy theory generally.The author emphasizes the geometric aspects of the subject, which helps students gain intuition.A unique feature is the inclusion of many optional topics not usually part of a first course due to time constraints: Bockstein and transfer homomorphisms, direct and inverse limits, H-spaces and Hopf algebras, the Brown representability theorem, the James reduced product, the Dold-Thom theorem, and Steenrod squares and powers. ... Read more

Customer Reviews (19)

Terrible textbook
This book is horrible if regarded as a mathematics book. Like previous reviewers I feel there is a total lack of clarity and rigor. Definitions are lacking, perhaps the author feels it is better to provide a "intuitive" feel for the material, than just definingg things. He fails miserably. The fact that what we are really dealing with in this subject are functors((co)homology, homotopy ) is nearly absent from the text. Instead drawings and pictures that are meant to provide "geometric" feel are supplanted.
I would state that this book attempts to teach how to compute in and use the theory than have you understand how the theory is built. It is a book for using the oven, not understanding how it works.

More Hand-Waving Than an Orchestral Conductor
In the TV series "Babylon 5" the Minbari had a saying: "Faith manages."If you are willing to take many small, some medium and a few very substantial details on faith, you will find Hatcher an agreeable fellow to hang out with in the pub and talk beer-coaster mathematics, you will be happy taking a picture as a proof, and you will have no qualms with tossing around words like "attach", "collapse", "twist", "embed", "identify", "glue" and so on as if making macaroni art.

To be sure, the book bills itself as being "geometrically flavored", which over the years I have gathered is code in the mathematical community for there being a lot of cavalier hand-waving and prose that reads more like instructions for building a kite than the logical discourse of serious mathematics.Some folks really like that kind of stuff, I guess (judging from other reviews).Professors do, because they already know their stuff so the wand-waving doesn't bother them any more than it would bother the faculty at Hogwarts.When it comes to Hatcher some students do as well, I think because so often Hatcher's style of proof is similar to that of an undergrad:inconvenient details just "disappear" by the wayside if they're even brought up at all, and every other sentence features a leap in logic or an unremarked gap in reasoning that facilitates completion of an assignment by the due date.

Some will say this is a book for mature math students, so any gaps should be filled in by the reader en route with pen and paper.I concede this, but only to a point.The gaps here are so numerous that, to fill them all in, a reader would be spending a couple of days on each page of prose.It is not realistic.Some have charged that this text reads like a pop science book, while others have said it is extremely difficult.Both charges are true.Never have I encountered such rigorous beer-coaster explanations of mathematical concepts.Yet this book seems to get a free ride with many reviewers, I think because it is offered for free.In the final analysis is it a good book or a bad book?Well, it depends on your background, what you hope to gain from it, how much time you have, and (if your available time is not measured in years) how willing you are to take many things on faith as you press forward through homology, cohomology and homotopy theory.

First, one year of graduate algebra is not enough, you should take two. Otherwise while you may be able to fool yourself and even your professor into thinking you know what the hell is going on, you won't really.Not right away.Ignore this admonishment only if you enjoy applying chaos theory to your learning regimen.

Second, you better have a well-stocked library nearby, because as others have observed Hatcher rarely descends from his cloud city of lens spaces, mind-boggling torus knots and pathological horned spheres to answer the prayers of mortals to provide clear definitions of the terms he is using.Sometimes when the definition of a term is supplied (such as for "open simplex"), it will be immediately abused and applied to other things without comment that are not really the same thing (such as happens with "open simplex") -- thus causing countless hours of needless confusion.

Third: yes, the diagram is commutative.Believe it.It just is.Hatcher will not explain why, so make the best of it by turning it into a drinking game.The more shots you take, the easier things are to accept.

In terms of notation, if A is a subspace of X, Hatcher just assumes in Chapter 0 that you know what X/A is supposed to mean (the cryptic mutterings in the user-hostile language of CW complexes on page 8 don't help).It flummoxed me for a long while.The books I learned my point-set topology and modern algebra from did not prepare me for this "expanded" use of the notation usually reserved for quotient groups and the like.Munkres does not use it.Massey does not use it.No other topology text I got my hands on uses it.How did I figure it out?Wikipedia.Now that's just sad.Like I said earlier:one year of algebra won't necessarily prepare you for these routine abuses by the pros; you'll need two, or else tons of free time.

Now, there are usually a lot of examples in each section of the text, but only a small minority of them actually help illuminate the central concepts.Many are pathological, being either extremely convoluted or torturously long-winded -- they usually can be safely skipped.

One specific gripe. The development of the Mayer-Vietoris sequence in homology is shoddy.It's then followed by Example 2.46, which is trivial and uncovers nothing new, and then Example 2.47, which is flimsy because it begins with the wisdom of the burning bush: "We can decompose the Klein bottle as the union of two Mobius bands glued together by a homeomorphism between their boundary circles." Oh really?(Cue clapping back-up chorus: "Well, ya gotta have faith...")That's the end of the "useful" examples at the Church of Hatcher on this important topic.

Another gripe. The write-up for delta-complexes is absolutely abominable. There is not a SINGLE EXAMPLE illustrating a delta-complex structure.No, the pictures on p. 102 don't cut it -- I'm talking about the definition as given at the bottom of p. 103.A delta-complex is a collection of maps, but never once is this idea explicitly developed.

A final gripe.The definition of the suspension of a map...?Anyone?Lip service is paid on page 9, but an explicit definition isn't actually in evidence.I have no bloody idea what "the quotient map of fx1" is supposed to mean. I can make a good guess, but it would only be a guess. Here's an idea for the 2nd edition, Allen: Sf([x,t]) := [f(x),t]. This is called an explicit definition, and if it had been included in the text it would have saved me half an hour of aggravation that, once again, only ended with Wikipedia.

But still, at the end of the day, even though it's often the case that when I add the details to a one page proof by Hatcher it becomes a five page proof (such as for Theorem 2.27 -- singular and simplicial homology groups of delta-complexes are isomorphic), I have to grant that Hatcher does leave just enough breadcrumbs to enable me to figure things out on my own if given enough time.I took one course that used this text and it was hell, but now I'm studying it on my own at a more leisurely pace.It's so worn from use it's falling apart.Another good thing about the book is that it doesn't muck up the gears with pervasive category theory, which in my opinion serves no use whatsoever at this level (and I swear it seems many books cram ad hoc category crapola into their treatments just for the sake of looking cool and sophisticated).My recommendation for a 2nd edition:throw out half of the "additional topics" and for the core material increase attention to detail by 50%.Oh, and rewrite Chapter 0 entirely. Less geometry, more algebra.

A final note:in actual fact I think of this book as rating around 3.4 stars, so it becomes a victim of rounding here.

Really bad as a "readable" texbookbut good reference
I am not able to understad why people seems to love this book my feelings, beeing mixed, are perhaps closer to hate.

The book is OK if (and only if) you previously know the matter but the lack of clear definitions, the excessive reliance in reader geometrical intuition, the conversational style of demos the long paragraphs describing obscure geometric objects, etc make it very difficult to follow if it is your first approach to AT.

On the other hand has useful insigths if you already know the matter.

If the purpouse of the author has really been to write a "readable" book (as he told us repeatedly) I think the attemp is a complete failure.

On the other handthe "Table of contents" is excellent and is a very good book for teachers,I think this is the reason of itspopularity.

If you can afford the cost, purchase J Rotman "An introduction to Algebraic Topology" and you really will get a "readable" book

amazing book, but caveat emptor
I think that Allen Hatcher has given us all something very valuable in this book.If you are like me, you've had those moments when reading in a math book when you read a sentence, and your eyes shoot open and you suddenly feel like someone has been standing behind you that you never knew was there.There are lots of those kinds of sentences in this book.On the other hand, I view it as a supplement to a book like Munkres or Bredon that provides the rigor necessary to allow the learner to figure out the topologist's geometric language.I have used these three and found them to compliment one another well.

excellent modern introduction
This is an excellent introduction to the subject. It's affordable, well-written, and the topics are well chosen. The presentation is modern, but includes enough intuition that the fairly naive reader (e.g., me) can see the point of things. I needed to (re)learn topology for a research project I was part of in the intersection of math/CS/statistics and this book was a big help. I wish that he had included simplicial sets in the topics, because I like the way he writes and would like to have a more elementary exposition tied to the rest of the book (I eventually found an expository paper that did a pretty good job, but worked out examples would still help with that topic), but it can't include everything. I highly recommend this book to anyone trying to get started in this fascinating subject. It will just scratch the surface, but it does a good job of that. ... Read more

Product Description
Combining concepts from topology and algorithms, this book delivers what its title promises: an introduction to the field of computational topology. Starting with motivating problems in both mathematics and computer science and building up from classic topics in geometric and algebraic topology, the third part of the text advances to persistent homology. This point of view is critically important in turning a mostly theoretical field of mathematics into one that is relevant to a multitude of disciplines in the sciences and engineering. The main approach is the discovery of topology through algorithms. The book is ideal for teaching a graduate or advanced undergraduate course in computational topology, as it develops all the background of both the mathematical and algorithmic aspects of the subject from first principles. Thus the text could serve equally well in a course taught in a mathematics department or computer science department. ... Read more

Product Description
Differential Topology provides an elementary and intuitive introduction to the study of smooth manifolds. In the years since its first publication, Guillemin and Pollack's book has become a standard text on the subject. It is a jewel of mathematical exposition, judiciously picking exactly the right mixture of detail and generality to display the richness within. The text is mostly self-contained, requiring only undergraduate analysis and linear algebra. By relying on a unifying idea--transversality--the authors are able to avoid the use of big machinery or ad hoc techniques to establish the main results. In this way, they present intelligent treatments of important theorems, such as the Lefschetz fixed-point theorem, the Poincaré-Hopf index theorem, and Stokes theorem. The book has a wealth of exercises of various types. Some are routine explorations of the main material. In others, the students are guided step-by-step through proofs of fundamental results, such as the Jordan-Brouwer separation theorem. An exercise section in Chapter 4 leads the student through a construction of de Rham cohomology and a proof of its homotopy invariance. The book is suitable for either an introductory graduate course or an advanced undergraduate course. ... Read more

Customer Reviews (15)

great old school (60's, 70's) math book
Back in the day there must have been a movement towards thin sleek books.Of course, there's tradeoffs.On the downside there's a lack of narration and context - the usual what, why, and where we're going type of stuff.The upside is what mathematicians call 'elegance'.For the layman this can be summarized as describing an object or thought in its most minimal form, whatever that may be.

My rating system of five stars is based on how successful the two authors succeed in the thin book paradigm.That is, I think there's enough there to "get it" and what's there is correct (or with a minimum of erratas).I agree with all the reviewers that gave this book 5 stars and 1-2 stars (it sucks).With respect to the 5 star folks, I agree that the authors met their objectives stated in the preface.They succeeded of creating an engineering schematic or wiring diagram to get from A to B.With respect with the 1-2 star folks I concur with their opinion.Yes, this book would not pass an editorial board standards based on modern publishing criteria.Nowadays it's more than just the handwaving.For example, reading chapter 1 section 1 I don't think there's enough there to explain to your grandmother what a manifold is and why should we care about them - a true test of mastery.

My advice to the 1-2 star folks that's not used to reading these thin sleek books is a technique I call pre-reading the book.It's a three-fold process.If you're forced into a quarter sytem (12-14 week) math class covering the whole book I highly recommend doing an 80-hour crash study session prior to the first class.Otherwise it'll seem like you just walked into a middle of a movie.First, read the preface to see where the authors 'think' they're going.Second, map out the key chapters and problems associated with each topical goal.Third, starting from the end of the book and going all the way to the start, build a dependency outline linking the 'big' result at the end with all the preceding.You'll be surprised how quickly you can cover what initially seemed advanced book.Most of the 'filler' can be gotten off the web.There's alot out there since this book has been used for the last 30 plus years.

My excitement from reading this book is how the authors bridge the study of smooth compact objects to topics that previously were in the realm of algebraic topology. These objects can be classified quickly by looking at global properties such as how many holes, described by the genus, how much curvature they have, described by the euclidean characteristic, how many vector zeros on the surfaces, described by how many bald spots would appear if hair sprouted on the surface and someone wanted to comb it and how many times the surface wraps around the interior, described by its degree.The authors do a good job of pretending like you don't have to know anything about algebraic topology but like I stated in the previous paragraph I couldn't resist googling because without getting some precursory knowledge it felt like being in the middle of a movie.

A spectacular book
I agree with the reviewer who is not a "higher mathematician". Neither am I; in fact, I repeatedly found that both Milnor and Hirsch became remarkably clearer after reading the same material from this book. So I stuck to this book. Chapter 4 is particularly well-written, with a very incisive discussion of connections among geometry, algebra, and topology. I hope the publishers decide to republish this book. How hard can that be in the modern small-volume printing era?

Poor beginning, good middle, ends as one long exercise.
I took differential topology as an undergraduate. We used this text. Neither I, nor any of the other students, had had any prior introduction to topological manifolds prior to taking this course. At that level, the problems with this book are immediate. The authors never define exactly what a manifold is. This is true of most mathematical objects introduced in the first and second chapters of the book. They are never precisely defined. Many of the exercises are very simple, testing your understanding of the definitions. But, without proper definitions, you are never sure what is being asked. You are also not sure what constitutes a correct answer. My feeling was that no one finished the class with any real understanding. A month after the class was over a professor asked me what a manifold was, and I couldn't answer him. I then took a course using Spivak's first volume differential geometry and a course in algebraic topology using Massey's book. With that background, I returned to this book and found it a delightful read up to the middle of chapter 3. Towards the end of chapter 3 the authors get totally lazy and make everything an exercise. That pretty much sums it up. If you have the mathematical background to consult other books for details, then you'll be able to get past the initially poor exposition and you'll find this book fun and more interesting as you get further into it. But, past a certain point, you'll realize that the textbook has become one long exercise or workbook, and not a textbook at all.

Good book
The book is very good, it is concise but I really like how the authors describe conceptual issues, it is easy to see the motivation behind most of the material we have covered so far, which was an issue in a book covering a similar topic I have read.

great introduction to the subject, despite its glaring faults
There are few books really suitable for undergraduates who wish to get a feel for differential topology, and among them Guillemin and Pollack is probably the best. Assuming only multivariate calculus, linear algebra, and some point-set topology (with a typical analysis class covering everything in the first and third categories), G&P presents an intuitive introduction to smooth manifolds with many pictures and simple examples while avoiding much of the formalism. It is most similar to Milnor's Topology from the Differentiable Viewpoint, upon which it was based, but it has additional material, most notably on differential forms and integration.

Books on differential topology (a.k.a. smooth manifolds or differential manifolds) tend to divide neatly into 2 types. Every book begins with basic definitions of smooth manifolds, tangent vectors and spaces, differentials/derivatives, immersions, embeddings, submersions, submanifolds, diffeomorphisms, and partitions of unity. Also the inverse function theorem is at least cited, if not proved (the proof is left to the reader here), as well as Sard's theorem and some sort of embedding theorem, usually Whitney's "easy" one. But beyond that the 2 types of books diverge, with one type treating vector bundles, the Frobenius theorem, differential forms, Stokes's theorem, and de Rham cohomology, and then possibly continuing on to differential geometry or Lie groups, such as in Lee's Introduction to Smooth Manifolds, Lang's Differential and Riemannian Manifolds, or Warner's Foundations of Differentiable Manifolds and Lie Groups, whereas the other type focuses on Morse theory, normal bundles, tubular neighborhoods, transversality, intersection theory, degree, the Hopf degree theorem, the Poincare-Hopf index theorem, and then possibly continues on to surgery, handlebodies, or cobordism, such as in Wallace's Differential Topology: First Steps, Milnor's TFDV, or Hirsch's Differential Topology. The first type of book is most suitable for the analytic aspects of the subject whereas the second for the topological, so comparing Lee to Hirsch is really an apples-to-oranges comparison. Mathematicians must know both, of course, but physicists, for example, usually use the first type more (although Dubrovin et al.'s second book, Modern Geometry. Part 2: The Geometry and Topology of Manifolds,is more of the second type). This book largely falls into the second category, but the final chapter covers differential forms, integration, Stokes's theorem, a little de Rham cohomology, and the Gauss-Bonnet theorem, making it somewhat of a hybrid, like Berger's Differential Geometry: Manifolds, Curves, and Surfaces (which focuses more on differential geometry) and Bredon's Topology and Geometry (which focuses more on algebraic topology).

The book is at its best when explaining concepts such as smoothness, transversality, stability, Whitney's theorem, intersection number, orientation, Lefschetz fixed point theorem, etc., pictorially, discussing the concept for a while before giving the definition or theorem. Many results that also can be proved using algebraic topology, such as the Brouwer fixed-point theorem, the Borsuk-Ulam theorem or the Jordan separation theorem, are proved, making the book much more interesting than those like Lee or Lang that just focus on machinery.

The main drawback of the book is its carelessness in definitions, particularly at the beginning. As some other reviewers have noted, manifolds are defined as being subsets of some Euclidean space, and diffeomorphisms are defined as being (a type of) maps between open sets in Euclidean spaces, which obviates an explanation of transition functions, but then makes awkward the many places later where references are made to "arbitrary" manifolds, which are never defined. But beyond the introduction, this embedding in some R^N is never used in the proofs, which use only local coordinate neighborhoods, so the results hold more generally (of course, every manifold does embed in some R^N, but one cannot use the proof of Whitney's theorem given here since manifolds were defined as subsets of Euclidean space to begin with).

The other notable feature of this book is its exercises, of which there are many, most being rather easy, but some being important theorems (the Whitney immersion theorem, smooth Urysohn theorem, tubular neighborhood theorem, etc.), with frequent hints provided. Later in the book, some proofs are rendered as extended problem sets, with the proof broken down into steps and each step treated as a separate exercise. This allows the reader to build up the ability to derive these results on his/her own, as well as forcing the reader to actually practice these techniques and thus truly learn the subject matter. ... Read more

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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
In this broad introduction to topology, the author searches for topological invariants of spaces, together with techniques for their calculating. 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 help students gain a thorough understanding of the subject. ... Read more

Customer Reviews (11)

Useless for learning OR a reference; disorganized; unfocused.
In my schooling for math, I have yet to encounter a worse text book than Armstrong.To begin with, the book opens with a long chapter that tries to motivate the subject by summarizing the rest of the book.Obviously this doesn't work out too well, as the reader has yet to even get a feel for topology; a lot of hand waving is utilized, and totally non-rigorous pseudo-definitions are given for important things, such as topological spaces themselves, that only serve to confuse later on.Then the author has the audacity to refer back to this chapter full of non-information when actually attempting to develop topology in a mathematically rigorous manner.

As for organization, there is none.The book buries theorems and proofs in paragraphs.There are no signals that proofs are over, and one normally has to search the chapters for relevant information.It is also worth noting that said information is often given in a most baffling order.

It's also important to point out that useful examples are almost nonexistent, and this is a major problem considering the level of exercises that Armstrong tries to give his readers.Speaking of the exercises, Armstrong often leaves very important theorems and definitions buried within these as well.And this is not your typical "leave to the exercises" complaint, oh no--he leaves incredibly important definitions and proofs to the reader, such as the existence of the one-point compactification.One may spend an entire class discussing this result, yet Armstrong leaves it to the student.

In the end, I would recommend this text to no one.Do not believe those who cite its mathematical "beauty."These people are fools.Get James Munkres's Topology 2nd Edition instead for your first course in Topology. For every terrible thing that I can say about Armstrong, I have a good comment about Munkres.An excellent alternative.

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. ... Read more

Product Description

Customer Reviews (12)

Topology Starter Book
I chanced upon this new arrival book in my local National Library book shelf 2 weeks ago. I congratulate the librarian who put this book at the public loanable section. Truely as the author claims, this Topology is for anyone with or without advanced math backgroud.
Topology has been the 'scarest' subject in the University where my 'sadistic' Math professors used it to 'kill' (to fail) students.
This book tells you Topology is fun and intuitive, it is a 'Rubber-sheet' Geometry as opposed to rigid objects in the axiomatic Euclidean Geometry.
Starting from the Euler formula "V-E+R = 2" (replace F = Face equivalently by R=Region,'V-E+R=2" easier remember as "VERsion 2" with a hyphen '-' before E), the central topic of Topology begins from Descartes / Euler to Riemann, Poincare...
The Epilog on "The Million-Dollar Question" (Poincare Conjecture) details the legendary Russian mathematician Perelman, who refused Fields Medal and in June 2010 rejected the Clay US$ 1 million prize.
Appendix A "Build your own Polyhedra and Surfaces", useful for students to make the 5 Plato Polyhedra on paper.
All in all, this book is excellent as a starter of Topology.

This book is a real gem itself
I just finished reading this book, and it only took me a couple of days. Admittedly, I have some exposition to math, having taken linear algebra, calculus, and differential equations, all of which are very useful in understanding this little book. But even a lay person with only some basic knowledge of geometry and algebra can grasp the content fairly easily. Most of the proofs are visual, and all of them are extremely elegant and simple. If you didn't take much math before but you are interested in the subject, get this book, and don't be scared of the apparent difficulty.

It is very rare that a math book is both so simple and so insightful. The topic is quite advanced, and the concepts, especially in the later chapters, are quite complex. And yet the author explains them in great simplicity. He doesn't go into some details I would have liked to see as someone with a math background, but that makes the book much more clear. And it's very well written. The author is very involved and obviously loves the subject. He also introduces other, related branches of mathematics like graph theory and knot theory, which could have made his book too complicated. And yet he deals with them so simply you might think that all math is like that. He presents all the beauty and elegance with little of the complexity which can make math seem so ugly and incomprehensible.

I really, really recommend this book, especially to the lay audience, and high school/undergraduate students in particular. Many of my friends like math but think that they are not smart enough. This book can show you that this is not the case, you just didn't think about it in the right way before. It's important that people stop thinking of math as something out of their reach, and all that is needed for that is a good teacher. And Richeson is certainly a good teacher.

That said, even a more advanced reader can enjoy this book, both for the incredible presentation, including many illustrations, and the elegant proofs and their sketches, which one can carry out to completion during leisure hours. The historical background is fascinating and the book reads almost like a novel.

The author did a really good job on this book, if only there were more books like it.

Very Good, But Challenging
Euler's Gem is a fascinating & well written book.However, it is also a pretty challenging read, one can not really sit back & read it straight through.But this is also what mathematics & learning is all about, as you often have to stop, re-read, & think a bit about what is being said.The claim is made that someone with only high school mathematics could read the book, & while this is probably true, it would be a steep climb.Especially as one progresses further & further into the book, many references are made to calculus, differential equations, & other related ideas, which the author does a fantastic job of explaining the ideas to people that never had the courses, but in the end it really would help the reader to have that knowledge beforehand.

What makes this a five star book is that it is so rich in knowledge.The average person won't be able to read it in a week, but if you're willing to put the time into the book, you'll get a lot of out it as it really is a great introduction to topology.Even if you can't pick up all the concepts, you're sure to be able to pick up many of the neat tricks the author points out, such as the wedding ring knot, coloring map problem, etc.Overall, one of the best books I've ever read, & one day I'll probably have to re-read it again because it's just so rich & packed with knowledge.

A Gem Indeed!
This is as good an introduction to topology as any for someone who isn't a professional mathematician. Even the professional can learn the history behind very familiar material.

A gem of mathematical results produced by one of the masters of mathematics
The title of the book is derived from the formula V - E + F = 2 that holds for any polyhedron. V is the number of vertices, E the number of edges and F the number of faces. First demonstrated by Euler, the proof of this result is surprisingly simple. As is the case with most such formulas and their proofs, there is at least one near miss in the history of mathematics. Descartes was close; in retrospect it is somewhat surprising that he didn't reach the appropriate conclusion. Of course, we are considering the great master Euler here, a giant of mathematics who was able to see things in his mathematical sight that people with the physical vision that he lacked overlooked.
Topology is a relatively recent area of mathematics, one of the few that can be considered to have had a point of origin and a creator. Richison works through the historical mathematical preliminaries of the formula, the shapes it describes were well known to the ancient Greeks yet they were nowhere close to the formula. Some historical and mathematical background on Euler follows this and it includes some of his other accomplishments. The last chapters describe some of the results that follow from topology in general and Euler's gem in particular. One of the most interesting is the theorem of combing a sphere, where the conclusion is that there must always be at least one hair that stands straight up. This may seem like an absurd thing for mathematicians to be concerned about but it has a major conclusion, that at all times there must be at least one point on Earth where there is no wind. Even more significantly it means that there will always be a zero.
Richison uses a large number of diagrams and formulas when needed, which is to his credit. Mathematics is based on equations so when an author deliberately avoids them in an attempt to increase sales, it is hard to claim that they are actually writing mathematics. This is an excellent book about a great man and a timeless formula. Well within the reach of the intelligent layperson, it is also a good book to use as a resource for a course where the students are required to make presentations.

Published in Journal of Recreational Mathematics, reprinted with permission.
... Read more

Product Description
This elegant book by distinguished mathematician John Milnor, provides a clear and succinct introduction to one of the most important subjects in modern mathematics. Beginning with basic concepts such as diffeomorphisms and smooth manifolds, he goes on to examine tangent spaces, oriented manifolds, and vector fields. Key concepts such as homotopy, the index number of a map, and the Pontryagin construction are discussed. The author presents proofs of Sard's theorem and the Hopf theorem. ... Read more

Customer Reviews (9)

it's ggrrrrrrrrrrreat!
I consider myself to be a pretty lousy graduate student and I still found this book to be very readable. This book is also cheap enough that you may want keep an extra copy around, as it makes a great gift item/stocking stuffer.

a must-read supplement for topology students
Milnor's "Topology from the Differentiable Viewpoint" is a brief sketch of differential topology, well written, as are all books by Milnor, with clear, concise explanations. For students who wish to learn the subject, it should be read as a companion to a more substantive text, such as Guillemin & Pollack's Differential Topology or Hirsch's Differential Topology, as too much of the material is left out for this to be adequate as a textbook. OTOH, it does make for good bedtime reading.

While this book is highly regarded among mathematicians, it is not without its faults, namely,
- it fails to cover many topics of importance, such as transversality (only mentioned in an exercise), embeddings, differential forms, integration, Morse theory, and the intersection form;
- it only cites some theorems without proving them, or it leaves the proofs to the reader;
- it offers proofs of many theorems that are really only sketches without all the details;
- manifolds are only defined as subsets of Euclidean spaces;
- there is only 1 collection of 17 problems at the end of the book, which are used to introduce important concepts; and
- it probably moves too quickly for true beginners, packing a lot into only 51 pages.

So don't buy this as your only, or even first, book on differential topology. Oddly, many of the faults that I listed above are simultaneously strengths, in that it can be read very quickly, with relatively little effort and a high rate of retention. Milnor really emphasizes the topology of the subject, giving applications such as the fundamental theorem of algebra, Brouwer's fixed point theorem, the hairy ball theorem, the Poincare-Hopf theorem, and Hopf's theorem. Most of the book focuses on degree theory, but there is also a nice introduction to framed cobordism, which is rare for an elementary book. Guillemin & Pollack's book was based in large part on this one, and could be read together, with G&P giving more elementary explanations and additional topics, while Milnor's book provides a proof of the Sard theorem and the Pontrjagin-Thom construction. The exercises, though not particularly difficult, do provide a good opportunity to practice proving theorems in the subject, as there are no hints for them, as one would find in many other differential topology books, and they are not separated by chapter.

Exactly would it should be
I would suggest to use this book as a companion to more serious books on topology. Weighing in at a mere 51 pages, this book accomplishes what it needs to: a brief, succinct introduction to topology mostly based on the work of Brouwer. There is a nice mixture of topics, ranging from Sard's theorem to Poincare-Hopf theorem. The proofs and ideas are not fully rigorous or developed, but that would be quite a bit to expect from such a short exposition.

best math book ever written

Despite the lovely subject matter covered in this book, it more importanty gives one a taste of Mathematics as an intellectual discipline. It in outline shows how a mathematical theory - in this case Differential Topology -is constructed and consquently what mathematicians actually do and think about.
Anyone who would like to appreciate Mathematics as a field of study rather than just learn some math should open this book.

Better still, the prerequisite is only multivariate calculus!I have long thought this book should be the third year of calculus rather than differential equations or complex analysis.

Additionally, for the novice it is the only entry I know of into the mysteries of high dimensional geometry, that amazing almost unbelieveable accomplishment of the human mind.

There is a Star Trek episode in which a blind woman wears a dress of sensors which enable her to know more about her environment than a person can know from seeing. She knows exact distances and dimensions, can detect minute movements, can process the complete spectrum of light. In some sense she sees better. Modern topology and geometry are like that sensor dress for seeing higher dimensions. While we can not visualize the sphere in 5 dimensions, we know more about it from these mathematical theories than a five dimensionally sighted being ever could.

Today, mathematics is often considered to be just a practical tool - like a spread sheet - or a toaster oven. We forget its power to widen our imagination, to frame the unimaginable. This book reminds us of this and shows why Mathematics is the Queen of Sciences.

Compact and useful
This book packs a lot of interesting material into a small volume. E.g., I picked up another book recently that started talking about cobordisms right off the bat; despite my having a couple of shelves full of well-known Dover, Springer, Cambridge UP etc. books on topology, differential geometry, mathematical physics, etc., Milnor's tiny book was the only one I found that could help me understand what cobordisms are right away. The book also uses many illustrations to help understanding.

I demote this to 4 stars only because Princeton UP's price is a bit high; many years ago I was lucky enough to find a used copy of the old U. Virginia edition, and paid much less. ... Read more

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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

Product Description

This book brings the most important aspects of modern topology within reach of a second-year undergraduate student. It successfully unites the most exciting aspects of modern topology with those that are most useful for research, leaving readers prepared and motivated for further study. Written from a thoroughly modern perspective, every topic is introduced with an explanation of why it is being studied, and a huge number of examples provide further motivation. The book is ideal for self-study and assumes only a familiarity with the notion of continuity and basic algebra.

Customer Reviews (4)

Best Intro to Topology
Topics are well motivated.
Theorems are proved in a rigorous yet intuitive style that one feels like it was an explanation rather than a dry proof typically found in the advanced math books.
Important key ideas are also sufficiently illustrated through examples and exercises.

If one finds it verbose, I'd recommendcroom--a bit more like the typical math books but accessible.

Surely not optimal
We used this book in an introductory topology class I took.Some of the exercises are poor (e.g. counting the number of topologies) and the exposition wasn't anything to go crazy about.After a while, I found myself reading Munkres exclusively; it's much more comprehensive.Maybe this book is well suited for folks looking to get a flavor of topology but nothing super concrete.

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

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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

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Customer Reviews (9)

If U want generalization out to infinity, this is it for you, in algebraic topology basics.
This former professor, and sometime head of the math department at U of Chicago, is quite a fellow. He is so DEEP that I many times didn't have a clue about some of his books. But this one seems more down at my level of intelligence, even though it is a whirlwind romance so to speak with algebraic topology basics.

Have at it, if you like the whirlwind!

The Title Says it All
I have always believed that the "goodness" of a mathematical textbook is inversely proportional to its length. J. P. May's book "A Concise Course in Algebraic Topology" is a superb demonstration of this. While the book is indeed extremely terse, it forces the reader to thoroughly internalize the concepts before moving on. Also, it presents results in their full generality, making it a helpful reference work.

The opposite of Hatcher
This book is clear, and direct.It tells you want you want to know.

Lucid and elegant, but not for beginners
This tiny textbook is well organized with an incredible amount of information. If you manage to read this, you will have much machinery of algebraic topology at hand. But, this book is not for you if you know practically nothing about the subject (hence four stars). I believe this work should be understood to have compiled "what topologists should know about algebraic topology" in a minimum number of pages.

A Unique and Necessary Book
Ones first exposure to algebraic topology should be a concrete and pictorial approach to gain a visual and combinatorial intuition for algebraic topology. It is really necessary to draw pictures of tori, see the holes, and then write down the chain complexes that compute them. Likewise, one should bang on the Serre Spectral Sequence with some concrete examples to learn the incredible computational powers of Algebraic Topology. There are many excellent and elementary introductions to Algebraic Topology of this type (I like Bott & Tu because of its quick introduction of spectral sequences and use of differential forms to bypass much homological algebra that is not instructive to the novice).

However, as Willard points out, mathematics is learned by successive approximation to the truth. As you becomes more mathematically sophisticated, you should relearn algebraic topology to understand it the way that working mathematicians do. Peter May's book is the only text that I know of that concisely presents the core concepts algebraic topology from a sophisticated abstract point of view. To make it even better, it is beautifully written and the pedagogy is excellent, as Peter May has been teaching and refining this course for decades. Every line has obviously been thought about carefully for correctness and clarity.

As an example, ones first exposure to singular homology should be concrete approach using singular chains, but this ultimately doesn't explain why many of the artificial-looking definitions of singular homology are the natural choices. In addition, this decidedly old-fashioned approach is hard to generalize to other combinatorial constructions.

Here is how the book does it: First, deduce the cellular homology of CW-complexes as an immediate consequence of the Eilenberg-Steenrod axioms. Considering how one can extend this to general topological spaces suggests that one approximate the space by a CW-complex. Realization of the total singular complex of the space as a CW-complex is a functorial CW-approximation of the space. As the total singular complex induces an equivalence of (weak) homotopy categories and homology is homotopy-invariant, it is natural to define the singular homology of the original space to be the homology of the total singular complex. Although sophisticated, this is a deeply instructive approach, because it shows that the natural combinatorial approximation to a space is its total singular complex in the category of simplicial sets, which lets you transport of combinatorial invariants such as homology of chain complexes. This approach is essential to modern homotopy theory. ... Read more

Product Description
This excellent introduction to topology eases first-year math students and general readers into the subject by surveying its concepts in a descriptive and intuitive way, attempting to build a bridge from the familiar concepts of geometry to the formalized study of topology. The first three chapters focus on congruence classes defined by transformations in real Euclidean space. As the number of permitted transformations increases, these classes become larger, and their common topological properties become intuitively clear. Chapters 4–12 give a largely intuitive presentation of selected topics. In the remaining five chapters, the author moves to a more conventional presentation of continuity, sets, functions, metric spaces, and topological spaces. Exercises and Problems. 101 black-and-white illustrations. 1974 edition.
... Read more

Customer Reviews (3)

Perfect book for the right reader
This book is not a textbook -- nor is it a rigorous buildup of topology from geometry. It is simple and conceptual bridge from the concepts of congruence classes in geometry, and basic ideas from map theory, into topology.
For any person wishing to acquire a grasp of the subject, I would recommend this book as a primer to any more conventional treatment of topology. In many texts, you are given general theorems in terms of sets, with no real idea of why these ideas have come about. From Geometry to Topology provides the necessary intuitive background... which is not to mention -- it's a darn interesting read.

Easy, but okay...
This book is quite pictorial and thus easy to read, as it's intended to be: a first introduction on undergraduate level to topology. However, it's a pity it doesn't go into topology very much. It stays very informal.

Very nice and intuitive introducation to topology
I bought this book shortly after my introductory analysis professor had once mentioned what is topology. That was the first topology book I had read, and I fell in love with the subject ever since. It's easy to understand,and very enlightening as well.
The book walks you through the transition from geometry to topology, then eleborates on several basic topological concepts.
Very interesting stuff ! ... Read more

Product Description
Contents: Introduction. - Fundamental Concepts. -Topological Vector Spaces.- The Quotient Topology. -Completion of Metric Spaces. - Homotopy. - The TwoCountability Axioms. - CW-Complexes. - Construction ofContinuous Functions on Topological Spaces. - CoveringSpaces. - The Theorem ofTychonoff. - Set Theory (by T.Br|cker). - References. - Table of Symbols. -Index. ... Read more

Customer Reviews (5)

great as motivation but not a textbook
While I agree with the other reviewers here that Jaenich's "Topology" is very well written, goes to great lengths to explain the "hows and whys" of topology, and includes many, many figures (about 1 per page on average), it is probably more popular with people who already know topology than with beginning students, even though it is an introductory text intended for undergraduates. This is due to both a frequent lack of precision or formality in proofs and definitions coupled with a tendency to discuss much more advanced material with which a student at this level wouldn't be familiar. I believe that experienced mathematicians, who perhaps learned point-set topology from books such as those of Munkres, Kelley, or Bredon (or even an analysis book such as Royden), appreciate how this book focuses on motivating the concepts, explaining how the various objects are used elsewhere in mathematics - for that purpose this is one of the finest books I have seen. However, too much material is mentioned that is certainly over the heads of most students new to topology, such as the Pontrjagin-Thom construction, the spectrum of commutative Banach algebras, or Lie groups, often in a very cursory manner that would serve only to confuse beginners. Concepts are often used before they are defined, or are not defined precisely, which is liable to frustrate these students as well. Many topics are given such short attention it makes you wonder why the author even bothered - such as a page devoted to Frechet spaces followed by a section consisting of a single paragraph on locally convex topological vector spaces. Much of the material is not covered very deeply - only a definition and maybe a theorem, which half the time isn't even proved but just cited.

Certainly this book couldn't be used as a textbook for an undergraduate course - for the reasons mentioned above and also because not enough material is actually covered, as well as the obvious deficiency in that it lacks exercises for the reader. Most of the proofs until the last chapters are of the 1- or 2-paragraph variety, with some pictures added, although as the book progresses the level becomes increasingly more sophisticated. The book also covers both point-set topology (topological spaces, compactness, connectedness, separation axioms, completeness, metric topology, TVS, quotient topology, countability, metrization, etc.) and elements of algebraic topology (homotopy, fundamental group, simplices, CW-complexes, covering spaces, but not really homology), but the presentation of the algebraic topology in particular is not liable to be helpful for the novice, except for the treatment of covering spaces, which is perhaps the highlight of the book. Half of the chapter on homotopy is actually concerned with categories and functors, probably not the best way to introduce the subject. In fact, here is direct quote from the index:
"We talk about homology (and a number of other objects beyond the realm of point-set topology) several times in this book, but the definition is not given."
That, in a nutshell, explains the difficulty with this book.

So why am I rating this 5 stars? For the wealth of examples (e.g., 4 sections on examples of quotient spaces) and explanations of how these concepts are used and why they are important. Just by looking at the contents one can see this, as there are sections titled:
"What is point-set topology about?," "What is algebraic topology?," "Homotopy - what for?," "The role of the countability axioms," "Why CW-complexes are more flexible," "Yes, but...?," "The role of covering spaces in mathematics," "What is it [Tychonoff's theorem] good for?"
The chapter on covering spaces, coming near the end of the book, is particularly good, with a proof of their classification given. This is definitely the most fleshed-out part; if only the rest of the book could go into this depth.

This book would make an excellent supplement to a more formal textbook such as Munkres, but is not a substitute for it. But I would still consider this as a must-read for all those students who plan on studying mathematics in graduate school.

A simple introduction to advanced mathematical concepts
This text gives the reason behind many advanced topological concepts and tantalizes the reader with it's varied applications.

Basic topological concepts of open, closed, continuous, product topology, connectedness,compactness and intro to separation axioms is presented in a logical concise and easy to understand way.

The author then delves into topological groups and vector spaces introducting Hilbert Banach and Frechet spaces ( albeit briefly ).

Quotient spaces,homotopy, complexes and urysohn and tietze lemma along with partitions of unity are tackled next.

I especially enjoyed the section on covering spaces with which it concludes.

Perhaps the single best accolade I can give the book is that it gives one inspiration and motivation to explore in greater detail mathematical objects discussed.

The text is useful to all students of mathematics and physics alike.

Full of motivations
This book is fun to read. In a weekly homework meeting for an Algebraic Geometry class, I complained to one grad student "Geometry textbooks should have many pictures", and he asked "Define 'many'?" I said "One on each page". Now this topology book is certainly close to that. (It has more than 180 illustrations.) Though its written style is a bit informal, 'handwaving' arguments can serve as outlines of rigorous proofs.

Since it does not have any problem sections, I can see why Munkres' book is more popular in college. It still gives some inspiring questions from time to time. Besides the basic pot-set topology, it also covers some algebraic topology and differentialtopology. The author does not hesitate to use examples from those advanced areas without formal definitions, and this was a bit annoying when I read it the first time. In this sense, the book is not really selfcontained. However, when finally a notion is formally defined, I can see it from various aspects in those examples. This really helps me understand topology better, and makes me want to explore them. After reading the existence thm of covering spaces in chapter 9, I realized that mathematics is really an art.

The index in the back of the book is in the format of short definitions, which can be used as a quick reference.

Students: BUY THIS BOOK!!!
It is not too often that a book about topology is written with the goal of actually explaining in detail what is going on behind the formalism. The author does a brilliant job of teaching the reader the essential concepts of point set topology, and the book is very fun to read. The reader will walk away with an appreciation of the idea that topology is just not abstract formalism, but has an underlying intuition that is rich in imagery. The author has a knack for allowing readers to "see into the future" of what kind of mathematics is waiting for them and how topology is indispensable in its study.

At the end of chapter three, which deals with the quotient topology, the author writes the following paragraph: "If is often said against intuitive, spatial argumentation that it is not really argumentation but just so much gesticulation - just 'handwaving'. Shall we then abandon all intuitive arguments? Certainly not. As long as it is backed by the gold standard of rigorous proofs, the paper money of gestures is an invaluable aid for quick communication and fast circulation of ideas. Long live handwaving!". This has to rank as one of the best paragraphs that has every appeared in a mathematics book, for it nicely summarizes the need for developing a feel for the concepts behind mathematics before moving on to the rigorous proofs. Physicists in particular, who must assimilate mathematics very quickly in order to apply it to real problems must have a pictorial, "playful" understanding of the mathematical constructions.

Thus the language that the author employs is informal, and a listing of the best discussions in the book would really entail a listing of every one in the book. There is not one part of the book that is not helpful or interesting, and the author delves into many different areas that involve the use of topology.

If you are a beginning student in mathematics, BUY AND STUDY THIS BOOK...BUY AND STUDY THIS BOOK. You will take away so much for the price paid.

Excellent
Excelent, clear, well-motivated introduction ... Read more

Customer Reviews (2)

Good introduction to the basics of topology
This book gave me my first introduction to the basic concepts of topology and I consider that to have been a point of good fortune. Everything is explained in detail at a level that is appropriate for people who have just mastered calculus. The authors were also thoughtful enough to have included solutions to the exercises, which is something that is just not done often enough. Coverage includes compactness, connectedness, mappings and fixed points; winding numbers, dividing pancakes and sandwiches and vector fields. When students ask me to recommend a basic book on these topics, this is the one that I recommend, which is the highest praise that I can give.

Great Introduction to Topology
When reading this book, I kept on wondering how good it will serve as the textbook for a semester-long high school intro to topology class! The authors placed great effort in making this book rigorous and rich in material yet at the same time very accessible (at least the first part) to the average high school junior or senior who's interested in higher math. The book builds up the fundamental concepts in general topology rather slowly to ease their digestion, and provides abundant examples along the way. Following the definitions and examples are celebrated theorems and their proofs that truly demonstrate the power and beauty of tology as well as mathematics in general. In fact, the whole book revolve around the "existence theorem" in one and two dimension (in one dimension, it's also known as the intermediate value theorem in calculus). This theorem is not only important in its own right, it is also intimately connected (not in the topological sense) with many concepts in topology. To prove the theorem for a disk in two dimension, the authors go through a thorough study of winding numbers and later on introduces vector fields, concept of homotopy, and interesting theorems like fixed-point theorem and ham-sandwich theorem. The later chapters of the book where these things are mentioned are rather obscure and difficult to understand, rather unlike the spirit of the earlier part; but by the time a high school senior gets to that point, he or she will probably be a mathematician enough to willingly dwell into these abstract wonders. ... Read more

Product Description
This material is intended to contribute to a wider appreciation of the mathematical words "continuity and linearity". The book's purpose is to illuminate the meanings of these words and their relation to each other. ... Read more

Customer Reviews (11)

Fantastically clear
I used this book recently for self-study. I found it to be very good despite the fact that it's relatively dated. The proofs are clear and the author attempts to inject intuition whenever possible. There are sections devoted primarily to motivating strange definitions or building up intuition about complicated theorems. There are also plenty of interesting (and doable) exercises.

Unfortunately, there are some omissions, for example filters are not mentioned and algebraic topology is completely avoided. Otherwise however, this book has aged very well and I commend the author's friendly and intuitive (yet completely rigorous) approach to the subject.

Great service!
The service overall was very good:

i) The item was as described, and
ii) It was shipped quickly

fantastic introduction to general topology
The first part of this book that deals with topology is a pedagogical masterpiece. After motivating the key concepts of compactness and continuity in the relatively concrete setting of metric spaces, the book goes on to abstract topological spaces, a beautiful section on compactness including the tychonoff theorem, and an extremely lucid development of the separation axioms and the proof of the urysohn imbedding theorem and the stone-cech compactification. I personally find the chapter on connectedness to be the weak link in this part of the book. Wherever possible, Simmons provides an exhaustive list of examples (especially when introducing the various types of spaces) that aids comprehension. Moreover, some of the central concepts (product topology) and deeper results such as the Stone-Cech compactification are easier to appreciate because the author has a section on topological properties of the relevant function spaces couple of chapters ahead and several exercises along the way. All in all, a highly recommended intro to the subject.

Didactic perfection
In the author's words in the preface, the dominant theme of this book is continuity and linearity, and its goal is to illuminate the meanings of these words and their relations to each other. The book, he says, belongs to the type of pure mathematics that is concerned with form and structure, and such a body of mathematics must be judged by its high aesthetic quality, and should exalt the mind of the reader.

The author's attitude can only be characterized as magnificent, and, if one is to judge his utterances in the preface by what is found after it, one will indeed find perfect evidence of his delight in mathematics and his high competence in elucidating very abstract concepts in topology and real analysis. Indeed, this has to be the best book ever written for mathematics at this level. It is a book that should be read by everyone that desires deep insights into modern real and functional analysis.

After a brief and informal overview of set theory, the author moves on to the theory of metric spaces in chapter 2. His emphasis is on the idea that metric spaces are easy to find, since every non-empty set has the discrete metric, and that metric spaces are good motivation for the more general idea of a topological space. The Cantor set, ubiquitous in measure theory, dynamical systems, and fractal geometry, is constructed as the most general closed set on the real line, i.e. one obtained by removing from the real line a countable disjoint class of open intervals. Continuity of mappings between metric spaces is defined, and also the concept of uniform continuity, the latter of which is motivated very nicely by the author. Then, the author takes the reader to a higher level of abstraction, wherein he asks the reader to consider all of the continuous functions on a metric space, and turn this collection into a metric space of a special type called a normed linear space, and, more specifically, a Banach space. Thus the author introduces the reader to the field of functional analysis.

A lengthy introduction to topological spaces follows in chapter 3. The author motivates well the idea of an open set, and shows that one could just as easily use closed sets as the fundamental concept in topology. And, most important for functional analysis, he introduces the weak topology, and shows how to obtain the weakest topology for a collection of mappings from a topological space to a collection of other topological spaces. The reader can see clearly that the weaker the topology on a space the harder it is for mappings to be continuous on the space.

Compactness, so essential in all areas of mathematics that make use of topology, is discussed in chapter 4. It is motivated by an abstraction of the Heine-Borel theorem from elementary real analysis, and the author shows how well-behaved things are on compact topological spaces. Some important theorems are proved in this chapter, namely Tychonoff's theorem, the Lebesgue covering lemma, and Ascoli's theorem.

Recognizing that the only functions able to be continuous on a space with the indiscrete topology are the constants, and that a space with the discrete topology has continuous functions in abundance, the author asks the reader to consider topologies that fall between these extremes, and this motivates the separation properties of topological spaces. Chapter 5 is an in-depth discussion of separation, and the reader again confronts function spaces, and their ability (or non-ability) to separate the points of a topological space. Spaces that allow such separation to occur are called completely regular, and this property has far-reaching consequences in analysis and other areas of mathematics. The Stone-Cech compactification is discussed as an imbedding theorem for completely regular spaces, analogous to one for normal spaces.

The intuitive idea of a space being connected is given rigorous treatment in chapter 6. Certain pathologies can of course arise when discussing connectedness, and the author shows this by discussing totally disconnected spaces, remarking that such spaces are very important in dimension theory and representation theory. Indeed, computational and fractal geometry is much harder to study because of the existence of these spaces.

Chapter 7 is important to all working in numerical analysis, wherein the author discusses approximation theory. The Weierstrass approximation and the Stone-Weierstrass theorems are discussed in detail.

A slight detour through algebra is given in chapter 8. Groups, rings, and fields are given a minimal treatment by the author, discussing only the basic rudiments that are needed to get through the rest of the book.

Banach spaces make their appearance in chapter 9, with the three pillars of the theory proven: the Hahn-Banach, the open mapping, and the uniform boundedness theorems. These theorems guarantee that the study of Banach spaces is worth doing, and that there are analogs of the finite dimensional theory in the (infinite)-dimensional context of Banach spaces. The theory of Banach spaces is very extensive, but this chapter gives a peek at this very interesting area of mathematics.

Banach spaces with an inner product are considered in chapter 10. These of course are the familiar Hilbert spaces, so important in physics and the subject of a huge amount of research in mathematics. The presence of the inner product allows constructions familiar from ordinary finite-dimensional vector spaces to carry over to the inifinite-dimensional setting, one example being the transpose of a matrix, which is replaced in the Hilbert space setting by a self-adjoint operator.

As a warm-up to the infinite-dimensional theory, finite-dimensional spectral theory is considered in chapter 11. The famous spectral theorem is proven. Then in chapter 12, the reader enters the world of "soft" analysis, wherein topological and algebraic constructions are used to study linear operators on spaces of infinite dimensions. Putting an algebraic structure on a Banach space gives a Banach algebra, and then the trick is deal with the spectrum of an element of this algebra. The reader can see the interplay between algebra, topology, and analysis in this chapter and the next one on commutative Banach algebras. Indeed, the Gelfand-Naimark theorem, that essentially states that elements of a commutative Banach *-algebra act like the functions on its maximal ideal space, has to rank as one of the most interesting results in the book, and indeed in all of mathematics.

Good Classical Introduction to Banach Algebras
This is a fine book, but not quite in the 5-star league. Let me elaborate. The book is divided into three parts: general topology, the theory of Banach and Hilbert spaces, and Banach algebras. The first two parts lead, by way of synthesis, to the last part, where some interesting but elementary results are proved about Banach algebras in general and C*-algebras in particular. I might mention, for example, the Spectral theorem for compact self-adjoint operators, the Stone representation theorem, and the Gelfand-Naimark theorem.

I can attest from personal experience that the book is well-written; indeed I worked through it chapter by chapter. But today there do exist a plethora of other treatments that can at least rival this text in lucidity, organisation and coverage. For example, for general topology, there is an excellent text by Willard titled 'General Topology',as well as Hocking and Young's old 'Topology'. Both of these go much further in the realm of point-set topology than Simmons. Similarly there are any number of well-written texts on functional analysis that cover the subject of Banach spaces, Hilbert spaces and self-adjoint operators very clearly. Indeed in some respects I feel the Simmons book was inadequate by itself and needed to be supplemented by a text on linear algebra; self-adjoint operators -- and by implication, the Spectral theorem -- need to be seen and manipulated in the finite-dimensional version before one examines their infinite-dimensional generalisation. The Simmons book is a bit weak here; one needs to be playing with matrices.

These are, however, minor quibbles. The book can be recommended to a junior- or senior-level undergraduate. ... Read more