Product Description
Basic treatment, specially designed for undergraduates, covers preliminaries (sets, relations, etc.), topological spaces, continuous functions (mappings) and homeomorphisms, special types of topological spaces, metric spaces, more. Geometric and axiomatic approach for easier accessibility. Exercises. Bibliography.
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Customer Reviews (2)

What you need to know
I am currently a Math Graduate Student who is primarily interested in algebra. While I own several books on algebraic topology and algebraic geometry, I was for a long time without a point set topology reference. This book tells me concisely what I need to know - and is an excellent bargain!

An unusual, and unusually good, book on topology
Most books on topology start with topology on the Rn and then introduce the finer points of topology. Baum's book starts right out with abstract point-set topology without skipping a beat. I learned general topology from this book and I'd absolutely recommend this to any student and instructor, along with Counterexamples In Topology by Steen and Seebach. ... Read more

Product Description
Like any books on a subject as vast as this, this book has to have a point-of-view to guide the selection of topics. Naber takes the view that the rekindled interest that mathematics and physics have shown in each other of late should be fostered, and that this is best accomplished by allowing them to cohabit. The book weaves together rudimentary notions from the classical gauge theory of physics with the topological and geometrical concepts that became the mathematical models of these notions. The reader is asked to join the author on some vague notion of what an electromagnetic field might be, to be willing to accept a few of the more elementary pronouncements of quantum mechanics, and to have a solid background in real analysis and linear algebra and some of the vocabulary of modern algebra. In return, the book offers an excursion that begins with the definition of a topological space and finds its way eventually to the moduli space of anti-self-dual SU(2) connections on S4 with instanton number -1. ... Read more

Customer Reviews (6)

An Introduction with Mathematical Integrity
Gregory Naber is to be commended for writing a thorough introduction to gauge field theory in which the mathematics is presented with clarity and rigor.For the professional mathematician who is interested in physics, or for the graduate student who prefers to see the mathematics "done right" in advanced applications to physics, Naber's wonderful two-volume set stands apart from its major competitors, nearly all of which were written by physicists, for physicists.

Despite the attention to mathematical rigor, it is clear that Naber intended his books to be accessible to a dual audience of physicists and mathematicians. For the physicists, he has included gentle introductory chapters on topological spaces, homotopy groups, principal bundles, manifolds and Lie groups, and differential forms. For mathematicians, the chapters on physical motivation, gauge fields and instantons, Yang-Mills-Higgs theory, Spinor structures, etc., provide unusually accessible introductions to some difficult physics materials.

Chapter 0 of the first volume is worth the price of both books, as it leads the reader, in 26 succinct pages, to a compelling appreciation ofthe natural "fit" of the Hopf Bundle to the task of providing a quantum mechanical analysis of the exterior of a single magnetic monopole.For outsiders who have become incredulous about the increasingly sophisticated uses of topology and geometry in theoretical physics, this example provides some much-needed assurance.As the reader quickly learns, the use of connections on principal fiber bundles is neither gratuitous nor mathematical overkill:indeed, the bundle machinery emerges quite NATURALLY as the simplest and best mathematical tool, perfectly fitted to the special problem at hand.

Any serious reader will want to buy both volumes of this set:Topology, Geometry, and Gauge Fields:Foundations (volume 1), and Topology, Geometry, and Gauge Fields:Interactions (volume 2).These books take their place alongside the work of authors such as Jerrold Marsden, Theodore Frankel, Barrett O'Neill, and Walter Thirring, all of whom write about modern mathematical physics in a way that does not obscure the true role of the mathematics.

correction to dost
The review "Easy reading, complete proofs, plenty of exercises, October 29, 2005 by Rehan Dost is of the first volume, Foundations, not this volume which is Interactions. Naber's books are crafted to bridge physics, undergraduate mathematics and graduate mathematics. This is one more of his beautiful volumes in applied mathematics.

Easy reading, complete proofs, plenty of exercises
This text is by far the best introductory text marrying basic concepts of physics with pure mathematics.

Some background in the basic concepts of vector calculus, linear algebra, complex numbers and group theory is required.

The author begins by motivating the mathematics by the pursuit of finding a vector potential to represent a magnetic monopole. We see that the topology of R3-0 precludes such a vector potential from existing. We see here a simple example of how the topology of a space affects the physics associated with it.
The importance of the vector potential as something other than a convenient computational tool is highlighted by a reference to essential inclusion in quantum mechanics. Thus we NEED such a potential.

The author now asks whether there is a "trick" or device to get around this difficulty. The device are principal bundles and connections. For example the potentials noted above must keep track of the phase of a charged test particle as it moves thru the field of a magnetic monopole. We need a "bundle" of circles ( representing the phase at each point ) over S2 ( the author explains why we need only consider S2 instead of R3-0, briefly we need only keep track of 2 of the 3 spherical co-ordinates ).
Thus a curve in S2 thought of as the particles trajectory will have to be "lifted" to the bundle space by a lifting procedure called a connection.
In a more general setting elementary particles have an internal structure ( spin etc ) which becomes apparent during interactions although may not be apparent in uniform motion thru a vacuum. Since the phase of the particle does not alter the modulus when calculating probabilities these do not change. However, when the particles interact phase differences are important. We need to keep track of such phases as the particles interact.

Thus we need a "bundle" over a 4-manifold ( keeps track of the particles space-time path ) to keep track of such internal states. One sees we also need a group to transform states into one another ( usually incorporated into the bundle ). Connections then model physical phenomena which mediate changes in the internal states.
We see that some connections satisfy the Yang-Mills equations and using the appropriate equivalence relation form Moduli spaces.

Now that may seem like alot to digest with only a spattering of mathematical maturity.

The beauty of the book is that the author starts from FIRST principles.

Chapter 1 introduces topological concepts of topology, continuity, quotient topology, projective spaces, compactness, connectivity, covering spaces and topological groups.

Chapter 2 introduces concepts of path lifting, fundamental groups, contractability, simple connectedness, covering homotopy theorem, higher homotopy groups

Chapter 3 introduces principle bundles, transition functions, bundle maps and principle bundles over spheres.

Chapter 4 introduces manifolds, derivatives on manifolds, tangent/cotangent spaces, submanifolds, vector fields, matrix lie groups, vector valued 1- forms, 2 forms and Riemann metrics

Chapter 5 gets to some physics with gauge fields and connections, curvature, Yang-Mills functional, moduli spaces, Hodge dual , matter fields and covariant derivatives.

At each step the author carefully provides complete proofs and easy exercises to ensure understanding.

It was a pleasure to read the book and complete the exercises. At no point did I feel frustration or boredom.

Don't waste your money
This review refers only to the book printing quality not to the contents.

I had purchased some books from Springer in the past (Like Arnold Mathematical Methods of Classical Mechanics, Lang Algebra etc..) and found them beautifully edited: good binding, paper etc..

And to my surprise I was very disappointed with the overall quality of this book, poor binding -glued instead of sewn- bad quality paper -forming waves at the binding spine, etc..

You pay for a quality item, a book you can use for years, and you get a hardbound crap that you can not left open in a table without holding it tight risking to lose the pages after a few days of use in the process.

I find this unacceptable in books costing 60$+. Sadly I find this to occur very often, publishers should be more careful with their printings and custumers should demand a better quality.

Don't waste your money.

A reader.

MATH AND TOPOLOGY
Topology is very important scince in the fields of mathematics. And it using in many of another sinceis. ... Read more

Product Description

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

The present book grew out of notes for an introductory topology course at the University of Alberta. It provides a concise introduction to set theoretic topology (and to a tiny little bit of algebraic topology). It is accessible to undergraduates from the second year on, but even beginning graduate students can benefit from some parts.

Great care has been devoted to the selection of examples that are not self-serving, but already accessible for students who have a background in calculus and elementary algebra, but not necessarily in real or complex analysis.

In some points, the book treats its material differently than other texts on the subject:

* Baire's theorem is derived from Bourbaki's Mittag-Leffler theorem;

* nets are used extensively, in particular for an intuitive proof of Tychonoff's theorem;

* a short and elegant, but little known proof for the Stone-Weierstrass theorem is given.

Customer Reviews (1)

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

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

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

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

Customer Reviews (6)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Product Description
This book is an introduction to elementary topology presented in an intuitive way, emphasizing the visual aspect. Examples of nontrivial and often unexpected topological phenomena acquaint the reader with the picturesque world of knots, links, vector fields, and two-dimensional surfaces. The book begins with definitions presented in a tangible and perceptible way, on an everyday level, and progressively makes them more precise and rigorous, eventually reaching the level of fairly sophisticated proofs. This allows meaningful problems to be tackled from the outset. Another unusual trait of this book is that it deals mainly with constructions and maps, rather than with proofs that certain maps and constructions do or do not exist. The numerous illustrations are an essential feature. The book is accessible not only to undergraduates but also to high school students and will interest any reader who has some feeling for the visual elegance of geometry and topology. ... Read more

Product Description
With this book and a square sheet of paper, the reader can make paper Klein bottles; then by intersecting or cutting the bottle, make Moebius strips. Conical Moebius strips, projective planes, the principle of map coloring, the classic problem of the Koenigsberg bridges and other aspects of topology are carefully and concisely illumined.
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Customer Reviews (2)

Great book for high school teachers and the popular audience
Written for the beginner in topology but still presupposing a certain amount of mathematical maturity, this book is great fun and should be useful to high school teachers, and those who might be giving public talks on mathematics to audiences who are very interested in mathematics, but who don't have a substantial background in the subject. It is always helpful in those scenarios to have concrete examples that will illustrate some of the more difficult constructions in topology.

In chapter 1, the author attempts to give an intuitive definition of topology. The author uses various pictures and handwaving arguments to explain various notions in topology, such as homeomorphism ("coffee cup = donut"), simply connected (object with no holes), homology (two circles can intersect at one point only), Jordan curves, and Euler's theorem.

In chapter 2, the author uses paper models to illustrate the topology of surfaces, both orientable and nonorientable. The author gives instructions on how to make a paper model of a Klein bottle, but cautions the reader that such a model is not an exact representation of the mathematical object rigorously defined in topology, since the surface passes through itself in the paper construction.

Chapter 3 is a set of instructions on how to make a "shortest" Moebius strip. The procedures for doing this are interesting and fun, for the author constructs a Moebius strip whose length is less than its width, by a factor of 1 over the square root of 3. He devotes an appendix for an improvement due to Martin Gardner of Scientific American fame.

In chapter 4 the author constructs what he calls a "conical Moebius strip". The author asks the reader to consider an annulus with a radial slit, with which of course one can construct a Moebius strip by twisting the ends and joining them. But he asks how large the hole must be in relation to the outside diameter. The answer to this is that one does not need any hole at all in order to carry out the construction. In fact, an angular segment can be cut out instead of the radial cut, and this leads him to construct the conical Moebius strip.

The author returns to the Klein bottle in chapter 5, and shows first what happens if the usual construction of the Klein bottle is cut down the center symmetrically: two Moebius strips are obtained. But to construct this model is difficult, so he gives alternate constructions for making the Klein bottle. He then shows what happens to the various models when the pieces are cut.

But how do you make a projective plane using scissors and paper? Intuitively one can imagine this would be very difficult, but the author shows ways to do it in chapter 6. His strategy for making these models is to teach the concept of symmetry in topology, and he pulls this off very well.

The famous 4-color problem for maps, i.e. that one needs only 4 colors for a map, is considered in chapter 7. At the time of publication, the 4-color problem was still open, so the author attempts to try and explain it using various diagrams and subdivisions thereof. The 4-color problem was proved using computer algorithms by the mathematicians K. Appel and W. Haken in 1976.

Network topology is considered in chapter 8, with the famous Koenigsberg bridge problem leading off the discussion. The author also introduces the very important Betti numbers, these having far-reaching ramifications in topology. And interestingly, the author is able, via a consideration of loop-cuts and cross-cuts in paper models of the Klein bottle and projective plane, to introduce the very important concept of "duality". The theory of knots makes its appearance here, although the discussion is very short. The author is well-aware of the difficulties in finding a classification theory of knots, but more could possibly be done here in the lines of the rest of the book to illustrate some of the peculiarities of knots.

Chapter 9 is really fun, for it concerns the torus with a puncture, and how to turn it inside out. The diagrams are helpful and the intuition gained valuable. The mathematician Steven Smale found a way of turning the sphere inside out in the early 1060s, but the author does not tackle Smale's method! This is unfortunate, since it is very difficult to follow the steps in Smale's method, at least for me.

The author does not want to leave the reader with the impression that topology is all scissors, paper, and tape, so he devotes the last two chapters of the book to point-set topology. Concepts such as continuity, limit points, and neighborhoods are discussed. It is quite difficult to explain to beginning readers and students of topology what a neighborhood actually does without having the notion of a metric or distance, but the author does a fairly good job here.

useful but not broad enough
The subject of topology lends itself to many different kinds of experimentation for undergraduate students.But this book spends a disproportionate amount of space on the Mobius strip and relatednon-orientability issues when it could deal with more knot theory andhomotopy theory than it does, and it could introduce finite topologies andMorse theory which abound in self-exploration. ... Read more

Product Description
Developed from a first-year graduate course in algebraic topology, this text is an informal introduction to some of the main ideas of contemporary homotopy and cohomology theory. The materials are structured around four core areas: de Rham theory, the Cech-de Rham complex, spectral sequences, and characteristic classes. By using the de Rham theory of differential forms as a prototype of cohomology, the machineries of algebraic topology are made easier to assimilate. With its stress on concreteness, motivation, and readability, this book is equally suitable for self-study and as a one-semester course in topology. ... Read more

Customer Reviews (7)

a masterpiece of exposition
This is a beautiful book which I have read and re-read with much profit and pleasure over the years.It presents topics in a very unusual order, which minimizes boring technicalities and develops intuition.Everything is very concrete and explicit, with lots of nice pictures and diagrams.

The book begins with a clear and concise treatment of deRham cohomology.If one hasn't seen differential forms before, then it might be a bit too brief and one might need to supplement it.But if one is comfortable with differential forms, then de Rham theory is a setting in which theorems such as Poincare duality can be proved with a minimum of pain.It is also very edifying to see the Poincare dual of a submanifold as a differential form.There is then a natural transition to Cech cohomology and double complexes.With this as a warmup, it is then a small additional step to spectral sequences (although the derived couple approach used here is perhaps not the most elementary possible).This machinery is then used to discuss an assortment of topics in homotopy theory and characteristic classes, which always sticks to the most important points without getting bogged down in technicalities.

It is highly unusual that the definition of singular homology only comes after the introduction of spectral sequences!This book might be best appreciated if one has some familiarity with singular homology and wants to better understand its geometric meaning.

Despite the avoidance of technicalities, the book is carefully written, although there is the occasional sign error.For example, the sign given for the Lefschetz fixed point theorem is wrong for odd-dimensional manifolds;try it for the circle and you will see.(Several other books make the same mistake.)

So far so good
I'm reading this book with my advisor.So far I've read through the first
five sections.My advisor is having me read this because he wanted me
to "read a really good book"So far I have no complaints. The arguments
are extremely clear and the book itself has a very smooth structure (no pun intended).

good book
It is a well written book. Useful for those whois learing algebric topology.

wonderfully clear, useful book
I agree with the other reviews, and only wanted to add to one of them that in regard to examples of chern classes, I believe they also use the whitney formula to derive the chern classes of a hypersurface from that of projective space, which really expands the realm of examples significantly.

This was all I needed in writing my notes on the Riemann Roch theorem for hypersurfaces in 3 and 4 space, for instance.I felt I knew little about concrete chern classes, but I was able to take the presentation in this book and use it for my purposes immediately.

A unique mathematics book
This book is almost unique among mathematics books in that it strives to ensure that you have the clearest picture possible of the topics under discussion.For example almost every text that discusses spectral sequences introduces them as a completely abstract machine that pumps out theorems in a mysterious way.But it turns out that all those maps actually have a clear meaning and Bott and Tu get right in there with clear diagrams showing exactly what those maps mean and where the generators of the various groups get mapped.It's clear enough that you can almost reach out and touch the things :-) And the same is true of all of the other constructions in the book - you always have a concrete example in mind with which to test out your understanding.

That makes this one of my all time favourite mathematics texts. ... Read more

Product Description
Differential geometry and topology have become essential tools for many theoretical physicists. In particular, they are indispensable in theoretical studies of condensed matter physics, gravity, and particle physics. Geometry, Topology and Physics, Second Edition introduces the ideas and techniques of differential geometry and topology at a level suitable for postgraduate students and researchers in these fields.

The second edition of this popular and established text incorporates a number of changes designed to meet the needs of the reader and reflect the development of the subject. The book features a considerably expanded first chapter, reviewing aspects of path integral quantization and gauge theories. Chapter 2 introduces the mathematical concepts of maps, vector spaces, and topology. The following chapters focus on more elaborate concepts in geometry and topology and discuss the application of these concepts to liquid crystals, superfluid helium, general relativity, and bosonic string theory. Later chapters unify geometry and topology, exploring fiber bundles, characteristic classes, and index theorems. New to this second edition is the proof of the index theorem in terms of supersymmetric quantum mechanics. The final two chapters are devoted to the most fascinating applications of geometry and topology in contemporary physics, namely the study of anomalies in gauge field theories and the analysis of Polakov's bosonic string theory from the geometrical point of view.

Geometry, Topology and Physics, Second Edition is an ideal introduction to differential geometry and topology for postgraduate students and researchers in theoretical and mathematical physics. ... Read more

Customer Reviews (13)

Excellent review of math for (particle) physicists
I bought this book to supplement my knowledge of mathematics which frequently is involved in understanding Particle physics concepts. The book is terse, but peppered with examples and insights about the definitions, and so far it is really fun to read. Seems like a good investment.

Too many errors to be useful for study
Reading all the glowing reviews of this book, I wonder whether the reviewers actually tried to use the book to understand the material, or just checked the table of contents. There are so many misprints, throughout, that one wonders if the book was proofread at all. Some of the mistakes will be obvious to every physicist - for example, one of the Maxwell equations on page 56 is wrong - others are subtle, and will confuse the reader. The careful reader, who wants to really understand the material and tries to fill in the details of some of the derivations, will waste a lot of time trying to derive results that have misprints from intermediate steps which have different misprints! Some chapters are worse than others, but the average density of misprints seems to be more than one per page.
The book might be useful as a list of topics and a "road map" to the literature prior to 2003, but that hardly justifies the cost (or the paper) of a whole book.

Geometry Topology and Physics: A condesed view
This book provide a complete and useful review of geometrical instuments of mathematical physics from the beginnig to the most advanced topics of interest. It can be used by students at the beginnig of thei studies in this topics, and it's found to be a useful gallery for higher level students (or scholar).

An excellent book
This is the best book of its type, that is, a book that contains almost all if not all the advance mathematics a theoretical physicist should know. I have studied chapters 2-9 and it has the perfect balance between rigorous presentation of topics and practical uses with examples. The level is for advance graduate students. The range of topics covered is wide including Topology topics like Homotopy, Homology, Cohomology theory and others like Manifolds, Riemannian Geometry, Complex Manifolds, Fibre Bundles and Characteristics Classes. I believe this book gives you a solid base in the modern mathematics that are being used among the physicists and mathematicians that you certainly may need to know and from where you will be in a position to further extent (if you wish) into more technical advanced mathematical books on specific topics, also it is self contained but the only shortcoming is that it brings not many exercises but still my advice, get it is a superb book!

A great reference book.
No doubt, the interplay of topology and physics has stimulated phenomenal research and breakthroughs in mathematics and physics alike.

Unfortunately, there is so much mathematics to master that the average graduate physics student is left bewildered.....until now.

The text is an excellent reference book. I emphasize reference. The book presupposes an acquaintance with basic undergraduate mathematics including linear algebra and vector analysis.

The author covers a wide range of topics from tensor analysis on manifolds to topology, fundamental groups, complex manifolds, differential geometry, fibre bundles etc.

The exposition in necessarily brief but the main theorems and IDEAS of each topic are presented with specific applications to physics. For example the use of differential geometry in general relativity and the use of principal bundles in gauge theories, etc.

Unfortunately, there are very few exercises necessitating the use of supplementary texts. However, to the author's credit appropriate supplementary texts are provided. The author goes to great lengths to show which texts inspired the chapters and follows the same line of presentation.

Perhaps the greatest attribute of the text is to take disparate branches of mathematics and coallate them under one text with applications to physics. In doing so one gains a better grasp of how the fields of mathematics interact in the domain of physics. ... Read more

Customer Reviews (4)

Not bad..
It's worth noting that there are quite a few in number of books out there on introductory (i.e. a first course in) alg. top.
In particular, I should mention that the book by Rotman and sizeable portions of Bredon, "Geometry and Topology" can serve as good supplementary reading. I still don't think \pi_1 should have been left out; although one *could* refer to theprequel,there's still more to be desired by way of completeness, if anything, as this book is intended for beginners. For instance, the relation between the fundamental group and the first homology group would have certainly shed some light on these seemingly (at first glance, anyway) disparateinvariants (as it isheavy-going onthe (co)homological apparatus altogether).

Munkres is byno means encyclopaedic, which is good, in opposition to, say, Spanier or Whitehead, and certainly warrants attention to worked-out examples in detail and some (not-so) routine exercises which makes this book accessible to wider mathematical audiences wishing to learn a little about this fascinating subject.

A little incomplete
This well written text is one of the standard references in algebraic topology courses because of its conciseness, and I find it very useful as a reference text.

However I think it is a little incomplete because of several reasons.

(1)It pays no attention to one basic concept ofalgebraic topology: the fundamental group.

(2) It doesn't cover ^Cechhomology, important in other areas, like dimension theory forexample.

(3) It doesn't stress the most important feature of algebraictopology: its connection to other areas of mathematics (analysis,differential geometry, etc.).

(4) Its list of references is too short,and lacks almost completely HISTORICAL references which are alwaysimportant to become an expert in any field.

Conclusion: a good referenceon homology and cohomology essentials, but not "the" reference onalgebraic topology as a whole.

The book binding is horrible
The material in the book (homology and cohomology theory, universal coefficient theorems, Kunneth theorem, duality in manifolds, applications to classical theorems of point-set topology) is for the most part solid. However...

- Munkres really belabors the simplicial theory, and it getsto be quite painful (especially the*CHAPTER* on the topologicalinvariance of simplicial homology groups).

- Some very important topics(homotopy theory, fiber bundles) are not at all discussed.

- The bookbinding is horrible -- my copy is in two pieces, with several loose pages,and I don't think the hardcover edition is still in print.

Excellent text on homology and cohomology
Algebraic topology is a tough subject to teach, and this book does a very good job.Some prerequisites, however, are essential:

* point set topology (e.g. in Munkres' Topology)

* Abstract algebra

* Mathematicalmaturity to be willing to follow a definition and argument even when itseems like a weird side-track

In addition, this would not be the firstbook I would recommend to those interested in algebraic topology.Firstmight be Massey's "Algebraic Topology: and Introduction" thatintroduces the fundamental group (conceptually easier than homology andcohomology).

At some point, however, a prospective student in topologywill have to learn homological algebra and this provides the most concreteapproach I know to the subject.

Algebraic topology is a lot of fun, butmany of the previous textbooks had not given that impression.This onedoes. ... Read more

Product Description
This book aims to provide undergraduates with an understanding of geometric topology. Topics covered include a sampling from point-set, geometric, and algebraic topology. The presentation is pragmatic, avoiding the famous pedagogical method "whereby one begins with the general and proceeds to the particular only after the student is too confused to understand it."

Exercises are an integral part of the text. Students taking the course should have some knowledge of linear algebra. An appendix provides a brief survey of the necessary background of group theory. ... Read more

Customer Reviews (4)

Not great but good
Let me begin by saying that I only got through the chapter on triangulations. I used this book in an undergraduate topology course where the instructor was not the best at teaching topology. So, I had to self-study with this book.

The first three chapters of the book are very well written. The theorems are proven in an intuitive manner that makes sense with some analysis background. Also, the exercises encourage, and at times force, the reader to really understand the topologies of the reals. The reader then is introduced to general topological spaces, including quotient and product topologies. This material is also very well written and relatively easy to understand, with some work by the reader.

The chapter on triangulations and surfaces is very difficult to understand. In the first few chapters, I could tell that the casual language chosen by the author would eventually lead to trouble. The careful word choices necessary in a math book were missing in this chapter. Little details like "relative to" and "in" are left out, sometimes requiring hours of careful reading of definitions trying to figure out exactly what the author means. This, to me, in unacceptable. The book reads more like lecture notes and less like a text book. Fortunately, I also purchased Munkres' topology book and referred to that whenever I didn't understand the author's explanation, which was a lot in the last chapter I studied.

Taking into account all the deficiencies with this book, I would still recommend it just for the first 3 chapters. These chapters are an excelent introduction to topology. I give this book 4 stars because it offers a good introduction to general topology. I also liked how the author put the exercises in the sections. This made it easy to see exactly what you should try to use in your proofs. I would also getting another, more theoretical, book to use as a reference if(when?) you get stuck by the author's poor choice of words.

The best undergrad topology text
Considering several other undergrad topology texts, e.g. Munkres, Armstrong, etc. this is the easiest to work with.Certainly the best text for self-study.The problems are not too difficult yet they help you grasp concepts as well.They are also laid out as you go; so every so often while you read the text you encounter a problem and you do it as you go.It is much better than putting them in the back of the chapters, as most text do.It is better to lay the problems in the text so you are encouraged to do them as you learn the material.The material in the text is very well explained and contrary to the previous review, is very well-suited, and with sufficient rigor, for mathematics students.The fact that this book "can be grasped at the sophomore level" as the previous revewer claims (and I agree with) lends credence to the simplicity of presentation of the material.Some reviewers I suppose aren't satisfied unless they see a hyperdense conglomeration of gobbledygook which characterizes so many mathematics texts.I don't fall into that camp and if you don't either and at the same want to begin study in topology then I highly recommend this book.

great Topology text
I must say that this is by far the best topology text that I have seen.Very readable, easy to follow.The anecdotal comments are also amusing.I'm particularly fond of the Ham and Cheese Sandwich Theorem!Highly recommend.

A very readable introduction to homology.
This is a very readable introduction to homology theory, replete with good illustrations and lucid writing.Kinsey does a great job of motivating and explaining her material.I only wish I could have had a text this readable when I was a student. ... Read more

Product Description
Concise work presents topological concepts in clear, elementary fashion without sacrificing their profundity or exactness. Author proceeds from basics of set-theoretic topology, through topological theorems and questions based on concept of the algebraic complex, to the concept of Betti groups.
... Read more

Customer Reviews (5)

Homology theory
This book introduces the algebraic machinery of homology theory and uses it to prove the invariance of dimension and the invariance of Betti numbers. With this point of view, manifolds should be taken to be complexes, and their properties should be studied in terms of linear combinations of its simplexes. So, for instance, in an appropriately oriented tetrahedron the "sum" of the faces is zero since each edge is counted twice with opposite orientation. But if we try to do the same thing in the projective plane or on a Möbius strip we will find that the summation of all faces leaves a boundary, revealing the difference between orientable and nonorientable surfaces. The natural algebraic equivalence of cycles of edges in a complex (homology) is close enough to topological equivalence (homotopy) to make the notion useful; in particular, the connectivity of a surface determines the number of generators of the free part of its homology group (the Betti number), so invariance of Betti numbers does give us useful topological information. Of course, if it was only for surfaces, homology and Betti numbers would be plainly inferior to the fundamental group, and indeed Alexandroff plays down surface topology while offering little in return except unsubstantiated reassurance that "anybody who wants to study topology for the sake of its applications must begin with the Betti groups".

A very bad topology book
When a book is translated it is supposed to be in English... this book is in topology as a language of it's own. For me with a library and long history of reading topology books, it is understandable. The author is recognized as a great mathematician, but I think he should have had his maiden aunt read his proofs for the book... he writes badly of topology and his references aren't anywhere near clear enough. I suppose it is that I found that his treatment of the tetrahedron and projective planereally don't work well when translated to numbers....

a mental roadmap
A very slender book, but it sets out the basic ideas of algebraic topology and HOW THEY RELATE TO EACH OTHER. A roadmap to what all this simplex stuff is all about.

For sophisticated mathematical readers only. Perfect adjunct to any first course.

(And a "lemniscate" is a figure 8).

A Gem
A great book, born in a great moment of mathematics. Alexandroff explains, and shows in pictures, what topology is basically about and why "homology groups" are the way to do it.

Anyone can follow this who has had multivariable calculus, plus seen the definition of a group (as in, say, arithmetic modulo 2).In 55 profusely illustrated yet rigorous pages Alexandroff shows how to define topological manifolds, cut them into "simplices", and keep track of simplices algebraically. He proves the two founding theorems of topology: the dimension of manifolds, and their homology groups, are both preserved by topological isomorphisms.

Alexandroff was a favorite student of Emmy Noether, and L.E.J. Brouwer, and followed Hilbert's lectures. The greatest algebraist, the greatest topologist, and the greatest mathematician of the early 20th century all had direct input into this book. All believed the most important, deepest mathematics can be made the clearest. They were right.

Excellent first exposure to algebraic topology.
This book is perfect for the advanced undergraduate--if you've taken modern algebra (groups, rings, fields, etc.), real analysis, some point-set topology, and are curious about algebraic topology, then this little book is time and money well spent. In about 50 relatively easy-to-read pages, you'll be able to sit down with your favorite topological spaces and actually do homological calculations. One of the book's main appeals is that it was originally published in 1932--well before homological algebra (aka "diagram chasing") obscured the beauty of algebraic topology. ... Read more

Product Description

Product Description

This book offers an introductory course in algebraic topology. Starting with general topology, it discusses differentiable manifolds, cohomology, products and duality, the fundamental group, homology theory, and homotopy theory.

From the reviews: "An interesting and original graduate text in topology and geometry...a good lecturer can use this text to create a fine course....A beginning graduate student can use this text to learn a great deal of mathematics."—-MATHEMATICAL REVIEWS

Customer Reviews (7)

never really liked it
but maybe you will.
i think it was Vick's book that i preferred.
Massey--I preferred Massey.
i think that this is the one for the grown-ups.

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

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

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

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

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

Customer Reviews (8)

Quite possibly the best text book written on any subject eve
I would give this book 10 stars if I could!

Many books, especially thosein the more theoretic regions of upper level mathematics are difficult toread, let alone use as a self-study text.The truth is that there is noreal need for an instructor when using this book.The only book that Ithink may be its equal or better is Griffith's book onElectro-Magnetics.

I have studied a great deal of mathematics, and I wishall the books I have laerned from (or tried!) were written HALF as well asthis one.

The content includes most, if not all, regions of fundamentalpoint-set topology.There is next to know differential or algebraictopology, but there are other texts for that.The illustrations areextremely helpful (and I am not even a visual learner!).It would bedifficult to give too much praise to this book, which is as comprehensiveas it is lucid.

Greatest math textbook I've read
This has to be the best textbook I've ever had for a class.Munkres is very clear and *detailed* in his proofs.Sadly, many authors skimp the details or brush aside technical difficulties, leaving the reader to fendfor themselves.Even in chapter 8, when he actually relaxes and does a few'picture proofs', he fills in more of the gaps than other authors, likeMassey, do when covering homotopy, fundamental group etc.His detailedproofs provide a good role model for when you're doing the exercises.

Butthis detail does not obfuscate matters.Munkres remains understandable. On the harder proofs he usually breaks things up into several steps, whichkeeps things readable.His examples are interesting, and his exercisesrange from easy to extremely difficult; actually most of them are of mediumdifficulty/somewhat hard variety.

I really feel that I'm getting a goodunderstanding of topology in my topology class, mainly because of thisbook.The challenging exercises give me confidence that my feeling isbased on some actual fact.All in all, a good experience.Hmmm...I guessI better finish reading the proof of the Jordan Curve Theorem.And getcracking on those homework problems.

great for independent studies
Ordinarily, independent studies in mathematics courses are difficult.I found that Munkres' book well explained the topics of point set topology (up to chapter 8) and algebraic topology (chapter 8.)His exercises, forthe most part, departed from the given examples and theorems, making themmore challenging, but doable.

Would be better iff....
I am a grad student and wow, this is unbelievable.Some of the notation is inconsistent and all the problems are difficult.I am taking the class during the summer and it is a 24-7 ordeal.

Excellent introduction: makes point set topology fun
This was my first introduction to point set topology as an undergraduate, and I enjoyed reading it even before taking the course.

Although not a hot research topic (compared to the rest of topology), it is foundationaland as such many have assumed that point set topology could only bepresented as a dull prerequisite for more interesting mathematics. Munkres' book, though, treats it as a goal of itself, as a fun world toplay in, and as such, has attracted many students to topology.

It isrecommended that a student first learn about metric spaces in a first-yearundergraduate analysis class before learning about point set topology. Although the material is self-contained, the motivations for thedefinitions are hard to understand without knowing the more mundaneexamples. ... Read more

Product Description
This book is intended to serve as a textbook for a course in algebraic topology at the beginning graduate level.The main topics covered are the classification of compact 2-manifolds, the fundamental group, covering spaces, singular homology theory, and singular cohomology theory.These topics are developed systematically, avoiding all unecessary definitions, terminology, and technical machinery.Wherever possible, the geometric motivation behind the various concepts is emphasized.The text consists of material from the first five chapters of the author's earlier book, ALGEBRAIC TOPOLOGY: AN INTRODUCTION (GTM 56), together with almost all of the now out-of-print SINGULAR HOMOLOGY THEORY (GTM 70).The material from the earlier books has been carefully revised, corrected, and brought up to date. ... Read more

Customer Reviews (2)

Good for newbies
Very nice way to start learning Alg topology. I am reading it for a class and it's been quite pleasant.

Excellent text on algebraic topology
The text contains material from the author's earlier two books; Algebraic Topology: An Introduction (GTM 56), and Singular Homology Theory (GTM 70). The book starts with an introductory chapter on 2-manifolds and thencontinues with the fundamental group; which is conceptually easier thanhomology, with which some books on algebraic topology start. The onlyprerequisite for this book is a basic knowledge of general topology; andthe book is easily accessible to anyone studying on his own. In short, Irecommend the book to anyone interested in algebraic topology. ... Read more

Product Description

Brings Readers Up to Speed in This Important and Rapidly Growing Area

Supported by many examples in mathematics, physics, economics, engineering, and other disciplines, Essentials of Topology with Applications provides a clear, insightful, and thorough introduction to the basics of modern topology. It presents the traditional concepts of topological space, open and closed sets, separation axioms, and more, along with applications of the ideas in Morse, manifold, homotopy, and homology theories.

After discussing the key ideas of topology, the author examines the more advanced topics of algebraic topology and manifold theory. He also explores meaningful applications in a number of areas, including the traveling salesman problem, digital imaging, mathematical economics, and dynamical systems. The appendices offer background material on logic, set theory, the properties of real numbers, the axiom of choice, and basic algebraic structures.

Taking a fresh and accessible approach to a venerable subject, this text provides excellent representations of topological ideas. It forms the foundation for further mathematical study in real analysis, abstract algebra, and beyond.

Customer Reviews (1)

Poorly conceived and executed
I was looking forward to reading this book and had high hopes that it would be a suitable text for our first year graduate course sequence in topology, since Professor Krantz has a reputation for being an excellent expositor of mathematics.

After going through about half the book however, I must say that I am seriously disappointed in this book. It seems to be poorly conceived and poorly executed.There are just too many topics covered in this book and most of them are covered very superficially.

Let me just give a small random sample of flaws I found.

1) The section of Morse theory in Chapter 2 is very cursory, hardly enough to give any useful insight into this topic. Moreover it refers to notions which are not defined (e.g. 1-cell, 2-cell) or to notions which are defined later (e.g. homotopy, defined in Chapter 3)
2) In Chapter 3 the author purports to cover singular homology theory without any mention of excision (or equivalently Mayer-Vietoris)!
3) In Chapter 4 the torus and Klein bottle are described as being obtained by pasting the sides of a rectangle.This however is never related to the notion of quotient space, which is treated very briefly in Chapter 2, and which notion is needed to rigorously describe what "pasting" really means.

This is by no means an exhaustive list, but should be enough to indicate the issues which plague this book.

I am not sure what audience this is aimed at.It seems to be a thoughtless jumble of formal arguments interspersed with a lot of informal "hand-waving".

So I am sorry to say that I can not recommend this book as a text in any of our courses.
... Read more

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

Customer Reviews (6)

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

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

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

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

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

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

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

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

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

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

Product Description
This book develops some of the extraordinary richness, beauty, and power of geometry in two and three dimensions, and the strong connection of geometry with topology. Hyperbolic geometry is the star. A strong effort has been made to convey not just denatured formal reasoning (definitions, theorems, and proofs), but a living feeling for the subject. There are many figures, examples, and exercises of varying difficulty. ... Read more

Customer Reviews (4)

a decent book, but
FAR worse than Thurston's classic notes, whence it came. Covers very little material by comparison, and has a lot more bogus "explanations". Given that the notes are available for free from Math. Sciences Research Inst. (MSRI), in convenient PDF form (for us Kindle DX Wireless Reading Device (9.7" Display, Global Wireless, Latest Generation)and Apple iPad MB292LL/A Tablet (16GB, Wifi) users, this is not a great book to get.

This book is the real deal by comparison
I have this book by Jeffery Weeks that Thurston's opens up with the real mathematics: The Shape of Space (Pure and Applied Mathematics). In both cases though, they fail to make the connection of the higher Cartan Lie Algebras to 3 manifold theory clear. When the connection of string theory to 3 dimensional geometry comes through this
approach to geometry, you think that we have a theory that neglects 5 dimensions and above?
So as good as this text is, it still falls somewhat short of what is needed in the modern world and we are forced to think of our own way to interpret the standard model symmetry breaking of SU(5) (CartanA_4) to U(1)*SU(2)*SU(3).

A refreshing style of writing
Stanislaw Ulam once compared learning mathematics to learning a language, in that some people learn mathematics by "grammar" while other learn it by ear.Thurston's book is a bit like learning by ear.

fun and geometric-intuition-minded
A must for anyone entering the field of three-dimensional topology and geometry.Most of it is about hyperbolic geometry, which is the biggest area of research in 3-d geometry and topology nowdays.

Most of it isreadable to undergraduates.Its target audience, though, is beginninggraduate students in mathematics.If not already familiar with hyperbolicgeometry, you might want to get an introduction to the subject first.Oncewith this background, though, you will discover there is another level ofunderstanding of hyperbolic space you never realized was possible.Oneimagines Thurston able to skateboard around hyperbolic space with the kindof geometric understanding he conveys here.

What made Thurston so famousand successful as a pioneer in 3-d topology and geometry was hisother-worldly geometric intuition.This book takes the reader along thefirst step of the 10000 miles of getting to that intuition. ... Read more

Product Description

Combinatorial algebraic topology is a fascinating and dynamic field at the crossroads of algebraic topology and discrete mathematics. This volume is the first comprehensive treatment of the subject in book form. The first part of the book constitutes a swift walk through the main tools of algebraic topology, including Stiefel-Whitney characteristic classes, which are needed for the later parts. Readers - graduate students and working mathematicians alike - will probably find particularly useful the second part, which contains an in-depth discussion of the major research techniques of combinatorial algebraic topology. Our presentation of standard topics is quite different from that of existing texts. In addition, several new themes, such as spectral sequences, are included. Although applications are sprinkled throughout the second part, they are principal focus of the third part, which is entirely devoted to developing the topological structure theory for graph homomorphisms. The main benefit for the reader will be the prospect of fairly quickly getting to the forefront of modern research in this active field.

Customer Reviews (1)

Help, where are the editors?
The book is poorly edited.
From the first chapter on, it is very difficult to decipher what the author is trying to say, because of linguistic and typographic errorsThere are evidently very interesting and useful things in this book, if you are interested in topology, homology and want to know about recent work on simplicial and other complexes, and especially if you are interested in applications to graph theory;but you have to be prepared to work very hard to find out what it is.With a basic introduction to simplicial homology at your side, the first few chapters should make sense. ... Read more