Topology Books

Regarding the content, I'm still reading the book but it is very interesting, I bought this book because it is a very good reference as far as the power supply design is concerned

Quick delivery. Quality product. Very informative. Very wide scope of design information, Very happy with purchase.

A very good book that covers most of the power converter architectures. An excellent addition to your personal library.

I recently graduated from Virginia Tech (undergrad), and I had only two classes that focused on power supply design/analysis. As an extension to what I learned in those classes (basics about buck/boost/flyback design and fabrication), this book is fantastic.

The book assumes you have a basic knowledge of EE principles, but nearly everything is explained in great detail. Topologies are examined one by one, and the author includes ALL of the derivations that lead to his design equations, which leaves very little room for misunderstanding. Each section contains pros/cons to using that particular topology, how to remedy common problems, and even talks a little about component selection (although since this book is years old, there are probably better components out there).

I haven't spent much time looking at the magnetics design section; however, it seems as though it would be useful. The chapter on loop compensation is excellent as well, offering a complete refresher of control theory and the design/analysis/use of Type 2 and 3 controllers. As I said before, the author assumes you're starting with very minimal knowledge of power supplies, so every equation and assumption is clearly justified in writing.

All in all, I would definitely recommend this text to anyone who is interested in power supply design or has to gain a quick understanding of something in the workplace since it not only includes the "quick and easy" design equations but also how to get there if you really care to know.

Although a good primer on the basic switching topologies, with an excellent chapter on inductor and transformer design, I couldn't help but feel that this book is more than a little outdated (which it is, at nine years old). There was no mention of synchronous or polyphase switchers, inductorless converters, charge pumps, high-frequency designs...and the section on MOSFETs left out what I feel was a great deal of information about paralleling and load sharing. Many of Linear Technology's app notes go above and beyond the material presented in this book...and they're free.

Simply, this book is fantastic! I am a graduate student in mathematics taking diff. top. and we are using this book. We have covered ch 1-7, 11-15 so far, and are now going back to finish immersion theory, with Milnor as a second reference. Most of this class I have taught myself the material directly out of Lee. The book is mind blowingly good for self study. I believe that an undergraduate with A. Calc and point set topology can work through this book and understand it well.

A few suggestions;

-worry less about the in-text exercises and more about the end-text exercises.
-go through the appendix before you begin, its easy and quick, but useful
-read with great detail. Lee provides everything, you may just not see it at first.

My field lies somewhere at the intersection of algebra, geometry and physics. This is a very handy reference, meaning a few pages accessible and contains most of the basic notions of differentiable manifold and some useful beyond-elementary topics for me. A good book to look into if you can't remember something in details.

This book is an antidote to the more common style of math text. So many math books feel like they were written by mathematicians, which is to say their authors prize being terse over being understandable. Far too many textbooks out there have me jumping on the internet to find explanations and examples of key concepts that absolutely must be understood completely for the sake of even being able to read the next ten pages. Lee spares his readers the trouble by taking things slowly and including all the steps they need to keep up with the discussion.

This was the textbook for my first class on smooth manifolds, and it is worth noting that it was the only class I took that semester in which I learned more from the textbook than from the professor's notes.

If you are a grad student, this book will make your life considerably easier.

By all accounts, this and Dr. Lee's other two books on manifolds are exceptionally well-written. But my copies arrived from Amazon this week, and, unfortunately, Amazon and Springer have decided to replace the crisp offset-printing of earlier printings by lower quality digitally-printed versions, probably as a cost-cutting measure.

If you care about how books look, I'd suggest trying Amazon marketplace or small retailers elsewhere to increase your odds of getting a superior copy from an earlier printing.

I should say first that I was already familiar with manifold theory before picking up this book. I had already wrestled with some of the definitions, theorems, and whatnot, so I can't necesarily say I was a complete beginner before reading this book. Also, I'm not sure if I can say how great this book would be if you have no idea what a manifold (or tangent space, etc.) is. However, that stuff aside, this is an amazing text. I'm studying this book on my own, and it's great. The concepts are woven throughout the text instead of being lumped into chapters devoted to them (though some people might prefer the latter). Also, they're used to reinforce and build on each other.

As an example, Spivak doesn't treat Lie groups until the second to last chapter. Lee introduces them in the second chapter, uses them as examples throughout the text, builds up the theory of Lie groups as the book goes on, uses Lie groups (and their actions on other manifolds) in developing certain other areas (it really streamlines the development) and ends with a nice big chapter on them. Of course, this is just one example.

Lee developes manifold theory so that it would appeal to a physicist, geometer, algebraist, topologist, etc. Everything gets talked about! This means, however, that he can't treat any one subject in too much detail. For instance, he leaves curvature and other parts of Riemannian geometry to his other Riemannian Geometry text, but it's definitely worth the trade off. This book trashes Spivak. Buy it today!

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.

I'd like very much this book. The book is very conceptual and also rigorous. It is self-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.

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.

Crystal clear in its structure, this book is my first introduction to topology, and I won't be needing another. If you like texts which are rather on the terse side, and don't mind doing exercises, then this is the book for you.

The only thing that bothered me at first was the lack of exercise solutions. But in retrospect, the time I spent thinking about hard exercises (which I would have looked up out of lazyness) turned out to be the most fruitful time of the entire reading experience.

This is a terrific introduction to topology. The problems are especially well chosen -- working through the problems will maximize your understanding. Competance in undergrad calculus is probably all that you'll need. Well written -- I can't imagine another book on the subject that would be more approachable.

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.

The service overall was very good:

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

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.

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.

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.

I can't give him five stars as good as Jeffery Weeks is, because the book doesn't even mentions Bianchi's manifold types. I also missed a discussion of his own manifold classification scheme and his program Snap Pea.
I bought the book so I could understand his low volume M003[3,-1] Weeks space. It ,too, isn't mentioned. He did give some coverage to the dodecahedron type of hyperbolic manifold.
He probably should have left cosmology out as he seems to have very little idea of particle formation during the early universe?
There is no doubt that Jeffery Weeks is a brilliant geometer, he just seems to have limited his background not to include fractal/ scaling theory, gravitational physics or Lie algebras?
What makes this book so good is his coverage of orientability in Manifolds. The only real physical evidence that nature may have orientablity incorporated is the parity of the electron, some asymmetry in mesons and times arrow in thermodynamics. His ideas and the dialog in the text about the expansion of space and what there was before the big bang is sophistry ( that is the bad teaching before philosophy)
and not science or mathematics. That he presents it is a shame on his thesis adviser who was supposed to have made sure he knew what meta-mathematics and metaphysics were?
I think it is a well written book and should be read by
those wishing to understand modern manifold theory,
it just isn't complete and he just assumes he knows much
more than is actually known.

This text is non-intimidating as an introduction to topology. Weeks carefully guides the reader through the building blocks of torii, Moebius strips, projective planes, and other surfaces. After working appropriate exercises, the reader gets a chance to visualize 3-manifolds and connected sums. Some aspects of these two topics can be difficult to explain, but analogies are applied to make understanding attainable. Further, figures and illustrations exist throughout the text, and these are definitely helpful for visualizing connected sums and non-orientable surfaces (both one-sided and two-sided).

(I especially like the approach to the Gauss-Bonet theorem using double lunes. It is a carefully crafted derivation with plenty of illustrations to avoid confusion.)

Some may think this text is too simple, but it is a "must read" for anyone who has not encountered topology and who wants to do individual research on the topic. Many texts claim to be introductory texts, but they are actually designed for those who already have a degree in math and who have seen similar subject matter. However, this one is definitely for "newbies." So don't worry.

This is a great book for anyone who is interest in Mathematical Topology and Cosmology Topology. This book does not require a reader to have strong mathematics knowledge. It only requires a reader to have patience to think and solve some problems in the book. The most brilliant point in this book is using diagrams to illustrate the Topology concepts, such as Manifold. This help the reader to get a "feeling" of some really difficult concepts in Topology. This book should be a classic like "Flatland".

chris tam
hong kong

I have a bachelors degree in Math.

As Feynman said, what we really mean by math is careful reasoning. This book brings you the joy of careful reasoning, guided by an expert.

Perhaps what turns some people off math in school is that the supreme example of careful reasoning is the mathematical PROOF. (Or perhaps it's just that most math teachers are so poor.) A proof tends to look dull and ponderous on the outside, and a student can easily miss the beauty of the underlying ideas. On the other hand, for your own amusement you can figure something out to your own satisfaction, without necessarily constructing a watertight proof. This book helps you do just that.

Many newspapers contain Sudoku problems, often with the reassuring claim that no math is required! People who hated math in school can be seen working happily on Sudoku puzzles, for the sheer joy of exercising their ability to reason carefully. The same ability would bring them far more joy while reading this book and answering the puzzles/exercises spinkled throughout.

This is a painless way to learn some advanced topology--or at least to gain insight. It's almost a picture book. Most problems include solutions and require only a few minutes of thought. They are also worth solving.

Now that I understand what is meant be a certain topology of the cosmos, I'm astounded that anybody actually considers it possible. Fascinating.

The only other books I have seen that deserve the title of both a reference and a textbook are by Lang. That being said, the exposition is at the other end of the spectrum with the goal being to teach the reader and not develop topology from the foundations in excruciating detail. The strength of the book (the same applies to the other titles by the same author) is that there is no compromise in rigor and the user-friendliness of the book relies upon the motivating discussions spread throughout the text. The author seems to have a long teaching experience and so he wisely advises the reader to work through the exercises. There is a pitfall with Lee's books: because the explanations are so lucid and intuitively satisfactory you might fool yourself that you know the material but this is not the case until you solve most of the problems.

I used Lee's 'Introduction to Smooth Manifolds' & 'Introduction to Curvature' for a few months, and I felt like it would be a good idea to complete the collection and acquire some more knowledge about topological manifolds using this book.

Overall, I think that Lee's 'Introduction to Topological Book' is an excellent book, as it is one of the few books that give both a profound intuition to the geometry of the subject, supplemented by rigorous proofs.
In my opinion, the book has only two disadvantages:
1) Not enough concrete examples.
2) It should has covered a bit more material, such as CW complexes.

All together, HIGHLY RECOMMENDED! The first chapter, giving some motivation to study manifolds is awesome!

I picked this book mainly because a friend recommended this whole series to me. While I cannot say this book would make a great introduction to point set topology (I think Munkres is still the best for that), it has all that one would want to get going with manifold theory. What I liked most about this text is probably the rigor. This text will motivate the topics and give rigorous proof to many theorems. There are also many good examples to illustrate his point. I'd recommend this book, and the follow-up text "Introduction to Smooth Manifolds" to anyone interested in graduate level mathematics. Since the two texts will likely cost you less than $100, they'll make a nice addition to your math library.

I began learning topology beyond real analysis with this book, and I found it to be a well-balanced text. This book covers every fundamental subject one needs to know without delving too much into a particular area of topology. The book begins with general topology and becomes increasingly algebraic as one progresses. Manifolds are emphasized throughout with ample examples and exercises. The presentation is very lucid and rigorous without being too pedantic.

There are more comprehensive books on topology, but this book is more apt for an introduction. I think that when one first learns about a mathematical subject, motivation is important. As a text goes deeper and deeper into the technicalities of a particular topic, the newcomer appreciates the concepts less and less and wonders where it is all leading to. This book affords just the right amount of material without causing one to reach the edge of boredom and lose sight of the bigger picture. In addition, a lot of motivation for learning the material is provided by the interspersed discussions on manifolds which are the most important topological spaces. The book prepares one for the entire field of topology in a concise manner.

Basic knowledge of metric spaces and group theory is recommended. If you are learning topology for the first time, you should definitely consider this book.

By all accounts, this and Dr. Lee's other two books on manifolds are exceptionally well-written. But my copies arrived from Amazon this week, and, unfortunately, Amazon and Springer have decided to replace the crisp offset-printing of earlier printings by lower quality digitally-printed versions, probably as a cost-cutting measure.

If you care about how books look, I'd suggest trying Amazon marketplace or small retailers elsewhere to increase your odds of getting a superior copy from an earlier printing.

Out of all the geometry books out there, this is one of the better.
If you must choose only one, this is a good choice.

Good luck,
Daniel

I have no experience with other geometry books--although I did use the Schaum book and other "outline" help books early in the school year as a reference. Actually Jacobs was easier to use than the "outline" help books. Many problems skate close to calculus (limits are introduced) and analytic geometry. Some problems are quite nearly elegant. Highly recommended.

As the other reviews show...this book is not only lucid and brilliant, but quite accessible. I've taught classes with it starting in Jr. High with gifted youngsters and with homeschoolers starting in the 6th grade.
I regularly recommend this book to homeschool moms because it is not intimidating, and they invariably enjoy it.
Also, this is PURE geometry, untainted by algebra. Probably the first and only time most students have to learn logic and the structure of argument.
A no brainer. This is the best geometry book I've ever seen, hands down.

A very clear, very entertaining textbook for a high-school course on geometry.

This book introduces logical proofs right at the beginning; you may have some difficulty convincing your kids or yourself that you need to work out all these silly logic puzzles in order to begin studying geometry, but you do.

From there on, the book is a sheer joy to read, full of interesting and tricky problems, clear explanations, and of course those famous B.C. and Peanuts clips.

This is the second book on geometry which I have read almost cover to cover. The first was Geometry by Ray Jurgensen and Richard G Brown written in 2000. Each of these texts seem to me to provide a good introduction to the basics of geometry. I suspect, even someone at the college level, can learn some items which could be quite useful for math, science, or engineering courses. The author has a wonderful sense of humor, which he springles over the text. I have not read the most recent edition of this book, but I hope to one day. This last edition is 17 years younger, having been published in 2004, instead of 1987.

This is an amazing book with an amazing subject (complex algebraic geometry). Every section presents something interesting and wonderful. I've only read chapters 0 (Complex manifolds, Hodge theory), 1 (Divisors & line bundles, vanishing theorems, embeddings), and 2 (Riemann surfaces). I had had a bad experience with alg geom before this book. Required reading for mathematicians in complex manifolds, algebraic geometry, or string theorists. There are some very trivial typos scattered, but nothing problematic in the least (like capital lambda instead of a big wedge, or indices). If you read the book carefully you will get a lot out of it.

I agree with most earlier commentators that this is a very nice introduction to the subject. That said, depending on your background, you may find that cover to cover may not be the most efficient way of reading this book. Also it differs from 'modern' treatments of the subject. All in all, it's an indispensible reference for most beginners and 'advanced beginners' if not more readers.

If you are a graduate student in mathematics or related fields and you are interested in learning algebraic geometry in the Griffiths-Harris way, then I suggest before buying this book to have a good background in the following:

1. Complex Analysis
2. Differential Geometry and calculus on manifolds
3. Homology-Cohomology Theory
4. Undergraduate Algebraic Geometry

Do not expect chapter 0, "Foundational Material", to be the place where you are supposed to build your "foundation". You can try the books of Michael Spivak, David A. Cox, Fangyang Zheng, among other books for foundational material but not chapter 0.

However, if you have most of the above-mentioned foundational material, then this book is good in presenting complex manifolds for example in chapter 0 section 2 and also in presenting (complex) holomorphic vector bundles, as well as many other things.

So, in summary, I would say a good book but not for students trying to learn the basics in algebraic geometry.

This book is fabulous - it is an indispensable reference for complex algebraic geometry. It is very clearly written and ideas are always motivated by examples and problems. Moreover, if you want to learn modern algebraic geometry, it's imperative to learn the classical case (over the complexes - which in practice is easier to work in) in order to understand the generalisations a la Grothendieck.

The book is beautifully written and easy to read, with emphasis on geometric picture instead of abstract nonsense. By far the best introduction to algebraic geometry for string theorists.

One of the best Topology books I have read. Even though the book has no figures (as one would expect from a topology book), almost every detail is covered and there are not obscure parts in the proofs. For example, the book by Willard is also good, but in some parts there are more complex details left for the reader. I took a basic topology graduate level course on the first half of 2007, which consisted on solving the problems in this book. We were able to find some problems that asked to prove something false, but they were three or four among all the problems from sections III to VIII. Anyway, this book is a classic that you should own if you plan to work in topology or at least read it while studying the subject. It's just a shame that the book is out of print.

This isn't the main focus of the book, but it does give a good introduction to function spaces and homotopy/paths, which are prerequisites to computational geometry/topology in computer science/economics modeling which is my field of work. It covers the fundamentals well, with narry a mistake that I can discover. There are so many writers who just throw something on the page, it's good to see someone who edits their own work.

Topology is about geography, shapes and sizes, folding and stretching/squeezing without breaking. Molding into the perfect shape for insertion into the environment.

True directions and purity of expression is what it's all about.

Dugundji does a great job of expressing these amazing concepts.
This is information that a surgeon would need as inspiration for his craft. Knowing what are the physical parts which make up something and how they fit together in close proximity is where it's at.

Metric spaces, convergence, function spaces, and completion are 4 things which do the trick of preparation for real, complex, and functional analysis at advanced levels. Many books of topology cover just that, but this one goes all the way to generalizations in other areas as is appropriate for topologists. Dugundji has a flair for organization which gives you things in a just-in-time delivery that keeps you motivated to the upteenth degree. Thanks to him for a job well done.

I like that this discusses sets, ordinals, and cardinals which are prerequisites to any mathematics. Then it covers the basics which are connectivity and compactness. Finally it gets tricky with identification and covering spaces. All right!

This book is clear, and direct. It tells you want you want to know.

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.

One of the reasons that Algebraic Topology is difficult to learn is that often the more general constructions (which are algebraic) are difficult to motivate visually. In fact, I have often found that attempts at visuallizing lead to confusion. J. Peter May avoids confusing illistrations in this book. Constructions are motivated by the results they consort. Most importantly May employes a thoroughly modern point of view. For example: the language of cofibrations/fibrations is used throughout, the handy idea of fundamental groupoid is introduced early in the treatment of the fundamental groups, there are a couple of chapters dedecated to homological algebra intersperced, both homology and cohomology are developed from the axiomatic point of view. May concludes the text with introductions to several more advanced topics such as cobordism, K-theory, and characteristic classes. The list of books that May offers in the suggestions for further reading section at the end is fairily comprehensive.

this is not a bad book, but it isnt for real. the back of the book says: ...treatment is sophisticated, no prior knowledge of the subject is assumed.

i think not.

you better be armed with a few other books and be prepared to spend some hours if you want to "learn" from this book as a beginner.

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.