Product Description

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

Customer Reviews (2)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

And this is where part 1 of this book ends.

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

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

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

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

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

thanks again:-) ... Read more

Product Description
Thorough, modern treatment, essentially from a homotopy theoretic viewpoint. Topics include homotopy and simplicial complexes, the fundamental group, homology theory, homotopy theory, homotopy groups and CW-Complexes and other topics. Each chapter contains exercises and suggestions for further reading. 1980 corrected edition.
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Customer Reviews (3)

Solid
A very interesting book which I enjoyed. I particularly found useful the "crash review" in algebra and analysis which functioned as a useful reference throughout the book.

Shouldn't be your first text in algebraic topology.
It is a decent book in algebraic topology, as a reference.At first, I found this textbook rather hard to read. Too manylemmas, theorems, etceteras. Three suggestions:

1. Needs more pictures, especially for the simplicialhomology Chapter.

2. CW complexes should be covered before duality and not after.

3. Needs more examples and exercises.

Overall, the book is very good, if you have already someexperience in Algebraic Topology. I found that the Croom'sbook "Basic concepts of Algebraic Topology" is an excellent first textbook. Too bad it is out of print, since it is very popular, every time I get it from the library, someone else recalls it. The combination of these two books probablyis the right thing to have: Maunder's book picks up whereCroom has left you.

Not bad.
Maunder's text may not be the "best" book on algebraic topology, but I still recommend this one to those who find other more advanced texts like Spanier rather inaccessible. Warning: the chapter on cohomology andduality is not very well-organaized (compared to other chapters), so youmay want to consult Bredon's book instead. ... Read more

Product Description

Customer Reviews (5)

Difficult to learn from.
As someone who came to this book with some exposure to analysis, multivariate calculus, and topology, I found it really difficult to learn from this book. I knew absolutely nothing about differential topology when starting, and cannot say I know much more now. The chapters on manifolds and submanifolds were easy enough to understand, but starting with the chapters tangent spaces and critical points, and critical and noncritical levels I just became lost. The big picture overviews given at the beginning of sections were somewhat helpful, but all in all I think I came away with very little from reading this book, considering the amount of effort I put into reading it. I found the book Introduction to Manifolds by Loring Tu to be much much better, although I've been told that it looks at things from more of a differential geometry point of view.

pictorial approach is great for total beginners, but lacks rigor
Wallace's "Differential Topology" is certainly the most elementary book on the subject that I've seen (and I've read dozens of such books). I wouldn't even say it is for "advanced undergraduates" - it could, and should, be read with only a background in multivariate calculus and basic linear algebra. It was intended to introduce the topological aspects of the subject without too much analytic or algebraic formalism, to build up a student's intuition. Technical details in this thin book are kept to a minimum and much of the presentation is done pictorially. Another notable feature is that it covers more advanced material, such as surgery, that most elementary books do not. However, due to a lack of rigor in some proofs as well as the limited range of topics covered, graduate students, and even senior undergrads who have studied topology, would be better served by higher-level introductory books, such as Guillemin & Pollack's Differential Topology, Milnor's Topology from the Differentiable Viewpoint or Collected Papers of John Milnor. Volume III: Differential Topology, or Broecker & Jaenich's Introduction to Differential Topology, although none of these books cover surgery. Another possibility is to read Gauld's, Differential Topology: An Introduction (Dover Books on Mathematics), which is a more advanced version of this book, but that has some problems of its own (cf. my review of it). This is not a textbook, but rather is designed for self-study; ideally it should be read as preparation for one of the above books or concurrently.

The presentation is so heavily weighted toward topology, there's no mention of analytical concepts such as differential forms, integration, metrics, vector bundles, or Lie groups, or even Sard's theorem or transversality, so don't expect this to be a substitute for Lee's Introduction to Smooth Manifolds, Lang's Differential and Riemannian Manifolds (Graduate Texts in Mathematics), or Barden & Thomas's An Introduction to Differential Manifolds. Instead Wallace introduces Morse theory and surgery (which he refers to by the alternative term "spherical modification"), and uses them to present a proof of the classification of 2-d surfaces that is different from the most common one based upon triangulations. He also includes standard results on embeddings, submanifolds, and immersions, as well as an introductory chapter on point-set topology for those who have no familiarity with the area. The amount of general topology covered is very small - only enough to define open sets, continuity, connectedness, and compactness - so if you haven't studied the subject already you'll need to learn it elsewhere, whereas if you have studied it, the first chapter could be skimmed or skipped.

The hallmark of this book is its informality, since the purpose of the book was to develop new students' topological and geometric intuition, before they become acquainted with more abstract concepts such as algebraic topology. Many of the proofs are not that rigorous, as steps are skipped and details omitted, and a number of important results are only cited or sketched. Sometimes pictures are relied upon for key steps in a proof, as most of proofs proceed by using embeddings in Euclidean space. This informality is both the biggest advantage of the book, as it can be read and grasped relatively quickly by even very inexperienced students, without getting bogged down in technicalities, and also its biggest weakness, since it is important for budding mathematicians to become proficient in proving theorems rigorously. At some point all the handwaving gets to be a little too much for my tastes.

One omission of this sort that I find particularly irritating is his almost complete failure to pay any attention to the differential structure of manifolds. Virtually never is a manifold actually shown to be smooth; even when a certain manipulation is asserted to produce homeomorphic manifolds, surgery is applied as if they were diffeomorphic. He practically uses the 2 words interchangeably, and doesn't even justify this by mentioning that all topological 2-manifolds can be smoothed. See Gauld for an example of how this detail should be handled properly.

In addition, some of the proofs use roundabout or inelegant methods, even when they are not necessarily easier to understand. For example, the proof of the existence of an embedding into Euclidean space for any manifold, rather than using the standard trick of having 2 nested sets of open coverings (which the author uses elsewhere), instead uses the explicit form of the bump function (the only proof the I can ever recall doing this), necessitating a tedious calculation, and embeds in a space that is of much, much higher dimension than necessary. His proof that an injective immersion of a compact manifold is an embedding is similarly convoluted and inefficient. Also, the proofs of cancellation of certain surgeries and rearrangement of surgeries are much harder to follow than the standard ones using handles or Morse theory, although in this case those given here at least have the benefit of being more elementary.

Oddly, for a book that has such a wealth of pictures and relies so much upon visualization, there are a couple of key points late in the book where a picture would have been a huge help, namely, on p. 106, when discussing rearrangements of surgeries, and especially on pp. 123-5, in the discussion of cancellation of surgeries. You will need to draw careful pictures to see what he is talking about.

There are exercises frequently sprinkled throughout the text that are used in fill in important missing steps in proofs, so doing them is really essential to learning the material. In fact, parts of the book just consist of a series of exercises with the author's guidance, such as the classification of non-orientable surfaces or the proof of the Morse lemma. However, I feel that some of them are perhaps demanding too much from beginners at this stage, as they are normally proved explicitly in even the more advanced books on the subject.

The are few mathematical typos, with only a couple being serious. On p. 124 it reads "phi and phi prime...respectively" when the author intends to say, "phi prime and phi in reverse...respectively." On p. 107 the number 0 is missing from the sentence "modifications of type on the 2-sphere" (they are of type 0). And most amusingly, on pp. 40-41, after constructing 2 open coverings, with the closure of one inside the other, the author explains that this is not an "unnecessary complication" but rather a "convenience," and then seemingly proceeds to not make use of this. However, he really is using it, but the reader cannot see that because in 3 places there is a prime on the variable U that should not be there.

In short, for students with virtually no experience with differential topology, this is a great place to start, but it is only a small "first step."

Your First Time
Wallace's is an ideal book for the budding mathematician with some interest in topology and familiarity with basic real analysis (e.g., Bartle). It takes the reader gently from first steps all the way through the complete classification of compact, smooth surfaces, with minimal fuss and bother in surprisingly few pages. (In other words, complete classification of these spaces is not as hard as one may have been led to believe elsewhere.) It is a truly wonderful book that a senior math major or beginning grad student can work their way through over a Christmas break, as I did and I hope they carry away the same fond memory that I do 35 years later.

a delight
deep mathematics made crystal clear and even elementary (to the senior college math major).

there are very few professional research mathematicians who write for beginners as does andrew wallace.i recommend all his books, although i have only read three of them, this one which classifies surfaces via morse theory, his intro to alg top via fundamental groups, and his other intro to alg top via covering spaces, classification of surfaces by triangulation, and fundamental groups

for those who do not know, morse theory is a beautiful and simple geometric theory that extends the second derivative test from calculus of two variables.think back at the picture of a surface in three space, the graph of a function of two variables, and recall the concept of a "level curve", or curve in the domain where the function is constant.

These level curves arise from passing a horizontal plane through the graph surface and projecting the intersection curve down to the x,y plane.In the case of a paraboloid, or bowl, graph of z = X^2 + Y^2, the curves look like circles or ellipses getting wider as you slice higher and higher.Thus the level curves down in the x,y plane form concentric closed curves.It is especially interesting that at the center, the level set is not a curve at all, but a single point, the minimum point of the graph.

If we consider a saddle surface, graph of Z = X^2 - Y^2, the slice by the horizontal plane through the origin is two lines, and all others, above and below, are hyperbolas.Thus again one can see from the geometry of the level curves, the geometry of the original graph surface.Here the second derivative test says there is no extremum.

We also know that for an infinite "trough" Z = X^2, in X,Y,Z space, the test fails, as any small perturbation can change the nature of the critical point at the origin.Morse theory says that, just as the second derivative test describes the shape of the graph at points where the second derivatives form an invertible matrix, so also the geometry of a surface can be reconstructed from the level curves of a single function defined on the surface, and having only such non degenerate critical points.

I.e. if at all critical points, the second derivative is non degenerate, then the geometry of the surface is entirely determined by knowing the index of the second derivative matrix at those critical points.E.g. a sphere is characterized by supporting a smooth function with exactly two critical points, one max and one min.

In between two successive critical points, the geometry of the surface does not change, and it looks like a "cylinder" i.e. a product of an interval with a single level curve. A torus, or surface of a doughnut, is characterized by having a function with one max, one min, and two saddle points.this is really making the solution theory of differential equations come alive and visible.

A quickie on differential topology
In this book, the author has given a quick taste of a very important subject, both in mathematics and in applications. Differential topology has found a niche in economics, physics, financial engineering, computer graphics, and computational biology, and it will no doubt find many more in years to come. It is also an area of mathematics that is still advancing, and there are many unsolved problems that can lead to interesting research programs. The author reviews elementary topology in the first chapter and then immediately introduces differentiable manifolds in the next. The presentation is very clear, and the author does not hesitate to use pictures to motivate and illustrate the main points. All of the discussion in these two chapters can be read easily by someone with a background in undergraduate calculus and some linear algebra. Special subsets of differentiable manifolds, the submanifolds, are considered in chapter 3, with the important embedding theorem proved. The theory of critical points follows in the next chapter. Although Morse theory is not mentioned, the author does define nondegenerate critical points, and shows, via a collection of exercises, the well-known result that a differentiable function in a neighborhood of such a point can be written as a quadratic form. A stronger embedding theorem is proven, namely one that allows an embedding of a compact manifold in such a way that the critical points are all nondegenerate. This discussion is generalized in the next chapter to critical and noncritical levels. The author motivates well the study of the neighborhood of a critical level by first discussing the properties of critical levels in the torus. The changing of the topology as one sweeps through the critical levels in this chapter is viewed as the process of spherical modification in the next one. The author does define what is meant by spherical modification, but does not use the usual terminology to discuss it, such as "cobordism" etc. he does however discuss the process of isotopy, and illustrates general position by means of intersections of curves. He illustrates these results in chapter 7 in the classification of two-dimensional manifolds. The usual proof is done in terms of simplicial complexes, but here the author proves it for differentiable 2-manifolds using critical point theory. The author ends the book by discussing how the subject could be pursued if the tools of algebraic topology were brought in. He discusses the killing of homotopy groups and motivates the theorem that an orientable compact 3-dimensional manifold can be obtained from a 3-sphere by cutting out a finite number of disjoint solid tori and filling the holes again with solid tori, with suitable identification of boundaries. He does not however prove when such constructions lead to the same 3-manifold, for this would lead to a resolution of the three-dimensional Poincare conjecture..... ... Read more

Product Description
Massey's well-known and popular text is designed to introduce advanced undergraduate or beginning graduate students to algebraic topology as painlessly as possible. The principal topics treated are 2-dimensional manifolds, the fundamental group, and covering spaces, plus the group theory needed in these topics. The only prerequisites are some group theory, such as that normally contained in an undergraduate algebra course on the junior/senior level, and a one-semester undergraduate course in general topology. From the reviews: "This book is highly recommended: as a textbook for a first course in algebraic topology and as a book for selfstudy. The spirit of algebraic topology and of good mathematics is present at every page of this almost perfect book." Bulletin de la Société Mathématique de Belgique#1 ... Read more

Customer Reviews (1)

one of the best books on algebraic topology
This is a charming book on algebraic topology.It doesnt teach homology or cohomology theory,still you can find in it:about the fundamental group, the action of the fundamental group on the universal cover (and the concept of the universal cover),the classification of surfaces and a beautifull chapter on free groups and the way it is related to Van-kampen theorem .After reading this book you will have a strong intuitive picture on "what is algebraic topology all about"(well at list on part of algebraic topology)read it an enjoy it!!!. ... Read more

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Customer Reviews (2)

Should've been better, but a bit too unorthodox
Gauld's "Differential Topology" is primarily a more advanced version of Wallace's Differential Topology: First Steps. The topics covered are almost identical, including an introduction to topology and the classification of smooth surfaces via surgery, and a few of the pictures and some of the terminology ("disconnecting surgery," "twisting surgery") are the same, too. But overall, Gauld is written at a higher level (even though it is also an introduction to the subject, for undergrads) and is much more rigorous. Also as with Wallace, the presentation focuses on topology, with no coverage of such analytic and geometric concepts such as Riemann metrics, differential forms, integration, Lie groups, etc., so this is not fungible with Lee's Introduction to Smooth Manifolds, Lang's Differential and Riemannian Manifolds, or Barden & Thomas's An Introduction to Differential Manifolds, in addition to being more elementary than these books. I probably would've rated this 5 stars if not for the unusual presentation of topological spaces and the sloppiness in the proof of the aforementioned classification.

The book begins with a 3-chapter introduction to point-set topology. Like Wallace, the amount of material covered is inadequate as a substitute for a course in general topology - only basic definitions (openness, neighborhoods, continuity) and Hausdorffness, connectedness, and compactness are covered. There is a decent amount of motivating discussion and examples, but this is done in by a serious flaw: The author uses his pet approach to topology via "nearness." Nearness is essentially the property of a point being an accumulation point (or limit point or point of closure - the terminology varies among authors) of a set. If one starts with a nearness relation, defined on the collection of points and subsets of a set, then one can define the concepts of neighborhood, openness, and continuity in terms of this relation. This is an idea that was pioneered in the '70s, first for the definition of continuity (which is particularly nice if one uses nearness) and then, by the author, for the definition of topology, where it is not as useful. This approach never caught on, so a beginning student who learns topology for the first time using this book may have problems when reading just about any other book on the subject; conversely, if that student has already learned topology elsewhere, this book will at first seem confusing. This aspect is probably the worst feature of this book.

After the brief introduction to topology, manifolds are defined and some useful analytic tools in R^n, such as the inverse function theorem, the Morse lemma, and bump functions, are treated. Another deficiency here is that manifolds are defined with no mention of the 2nd countability (or paracompactness) of the manifold. This isn't necessary for the purposes of this book since partitions of unity (and their applications) are never discussed and most of the theorems concern compact manifolds only, but the full definition should've been given for completeness (cf. Wallace, who does give the proper definition even though he, too, has no need for the 2nd countability). Morse theory is introduced fairly early in the book and then developed over the course of several chapters - the presentation is not as unified or complete as in Milnor's books, but is certainly superior to that of Wallace.

The book includes the standard proof of the easier Whitney embedding theorem (oddly, divided between chapters 7 and 15), as well as all 3 definitions of tangent spaces (similar to Broecker & Jaenich's Introduction to Differential Topology, but less concise) and some coverage of vector fields, albeit without precisely defining the tangent bundle (or mentioning vector bundles at all).

What really sets this book apart is the emphasis on differential structures and orientability via charts and bases of them. In other words, throughout the book, showing that a manifold (or map) is smooth or orientable, and comparing different smooth structure or orientations, is done by working with explicit choices of charts. It is this focus that is utterly lacking in Wallace. As an example, it is shown how to create infinitely many "different" differential structures on the real line by using different charts (but the author fails to note that these are all diffeomorphic). The many pictures of coordinate patches on manifolds and computations involving Jacobians really show the reader how one can deduce these properties in practice.

The treatment of surgery, from multiple perspectives, is also outstanding. First it is defined using embedded spheres and balls, and then the notion of differentiable gluing is used to improve the construction, to show that it is smooth (with no handwaving about smoothing corners). Next, the trace of the surgery is defined and that is related to the neighborhood of a critical point, using several different explicit representations of that neighborhood, with many excellent diagrams showing the different possibilities. Applying surgical techniques to classify surfaces starts well, even first using it to classify 1-manifolds, but the exposition starts to break down, with some results concerning moving critical points being proved in an informal manner, and a very confusing explanation accompanying an equally confusing diagram on page 200. Most disappointing is that Gauld substitutes homeomorphic for diffeomorphic in his ultimate theorem, but at least he (unlike Wallace) acknowledges this deficiency and points out that it can be overcome. After being so careful for most of the book, Chapter 14 should've been written at the same level of rigor.

The 2 appendices fill in some missing details in the proofs (Appendix A) and point toward extensions of the material (Appendix B). However, even in Appendix A, most of the proofs are just sketched. Moreover, and this is another irritating feature of the presentation, throughout the text a proof will sometimes just be given as one word: "Omitted." This means that the proof, or its sketch, is in Appendix A, but the author never tells the reader this. And then there are a number of places where a fact is used (such as that the genus of a surface is finite) but not justified, making it appear as if the author is making an unsupported assumption when in fact the claim is proved in the appendix; references to Appendix A really should've been added to the text. Appendix B touches upon other topics, such as other separation properties besides Hausdorffness, the classification of nonorientable surfaces (this should've been treated in the main body of the text, in more detail), Sard's theorem (not proved here), the smooth Brouwer fixed point theorem without homotopy theory, Hausdorff dimension (really out of place here), and the barest mention of dynamical systems. Most of this appendix is too little to be of any use.

This is a reprint of a camera-ready typescript (i.e., it looks like it was typed on an old typewriter), so it is hard on the eyes. Despite this, it is relatively free from typos, as well as other errors, until near the end of the book (e.g., in the last paragraph on p. 201 the variable S should have a tilde twice, on p. 202 the variable epsilon should've been defined before it was used, on p. 153 it should read "surgery of type (m,n)" not "to type (m,n)," on p. 112 the definition of U should include the words "range of h" not "domain of h" and the word "open" should've been inserted before "neighborhood" a few lines earlier).

There are a good amount of straightforward exercises following each chapter, with the results of some being cited in the text, so this could serve well as a textbook (it was designed for an experimental course), notwithstanding the issues that I've raised concerning some of its definitions. Don't be put off by the silly limericks that open the book, either. It's a shame - with a little more work and a little less idiosyncrasy, this could've been the best book on the differential topology for undergraduates, but even as is it makes a nice companion to Guillemin & Pollack's Differential Topology in that their overlap is not that large, so the 2 books combined do a good job of covering the subject.

Very good product
Nicely written, in understandable language, this book should stand amongst the references of its kind.

... Read more

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

Customer Reviews (1)

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

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

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

Product Description
Topology is a branch of mathematics packed with intriguing concepts, fascinating geometrical objects, and ingenious methods for studying them.The authors have written this textbook to make this material accessible to undergraduate students who may be at the beginning of their study of upper-level mathematics and who may not have covered the extensive prerequisites required for a traditionalcourse in topology. The approach is to cultivate the intuitive ideas of continuity, convergence, and connectedness so students can quickly delve into knot theory, the topology of surfaces, and three-dimensional manifolds, fixed points, and elementary homotopy theory.The fundamental concepts of point-set topology appear at the end of the book when students can see how this level of abstraction provides a sound logical basis for the geometrical ideas that have come before.This organization presentsstudents with the exciting geometrical ideas of topology now(!) rather than later.

Anyone using this book should have some exposure to the geometry of objects in higher-dimensional Euclidean spaces together with an appreciation of precise mathematical definitions and proofs.Multivariable calculus, linear algebra, and one further proof-oriented mathematics courses are suitable preparation. ... Read more

Customer Reviews (3)

Don't buy this book
This book is bad on many levels, possibly uncountably infinite levels. Nowhere in this book is there any motivation for the material, the author just throws various topics from topology at you and expects you to be interested, which I wasn't. The assignment's in the book are dull and boring and the entire book is entirely confusion. I find it slightly ironic that the book is called topology now, when every chapter I was waiting for the topology to begin. The only chapter in the book that is at least mediocre was the last chapter in the book. Other than that the author spends 25% of the book (right in the middle) discussing knot theory, a concept that he never relates to the rest of the book, or gives any motivation to learning he just throws it out there. Overall this book is too confusing for non-math majors and not rigorous enough for math-majors, so if you're looking for a good introductory topology book try Introduction to Topology by Colin Adams.

Product exactly as ordered, arrived quickly in great condition
What more can I say ... I ordered this for a friend who requested it, and who finds it invaluable.And it came very quickly.Can't do better than that.Sorry I'm not qualified to comment on the book's academic content.

Understandable is an understatement of the quality of the presentations
If you are looking for a text for an undergraduate course in topology, then this book is Goldilocks in disguise. The amount and level of the material are both just right. Chapter 1 begins the process by introducing topological equivalence and topological invariance. This is followed by chapters on knots and links, surfaces, three-dimensional manifolds, fixed points, the fundamental group and metric and topological spaces. The background mathematics needed to understand the contents of this book are all well within the skill set of an advanced undergraduate. There is the occasional appearance of a derivative, but an understanding of calculus is not needed.
The most significant skill is a through understanding of functions as mappings, and the special characteristics, such as homeomorphism, that functions can have. There are a large number of exercises at the end of the sections, further increasing its' value as a textbook. Topology is a branch of mathematics where one can sometimes engage in hands-on demonstrations. Problem 6 on page 38 is a demonstration involving cutting the toe off an old sock, sewing the ends together and then turning it inside out. Some of the other questions are a bit silly. The best is problem 6 on page 24, "Homeo, Homeo, wherefore art thou Homeo?"
The authors should be nominated for a prize in expository writing for this book, if it were in my power to do so I would. Understandable is an understatement of the quality of the explanations.

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

Product Description
This book introduces the important ideas of algebraic topology emphasizing the relation of these ideas with other areas of mathematics. Rather than choosing one point of view of modern topology (homotropy theory, axiomatic homology, or differential topology, say) the author concentrates on concrete problems in spaces with a few dimensions, introducing only as much algebraic machinery as necessary for the problems encountered. This makes it possible to see a wider variety of important features in the subject than is common in introductory texts; it is also in harmony with the historical development of the subject. The book is aimed at students who do not necessarily intend on specializing in algebraic topology.

The first part of the book emphasizes relations with calculus and uses these ideas to prove the Jordan curve theorem. The study of fundamental groups and covering spaces emphasizes group actions. A final section gives a taste of the generalization to higher dimensions. ... Read more

Customer Reviews (3)

A book of ideas
This book is an introduction to algebraic topology that is written by a master expositor. Many books on algebraic topology are written much too formally, and this makes the subject difficult to learn for students or maybe physicists who need insight, and not just functorial constructions, in order to learn or apply the subject. Anyone learning mathematics, and especially algebraic topology, must of course be expected to put careful thought into the task of learning. However, it does help to have diagrams, pictures, and a certain degree of handwaving to more greatly appreciate this subject.

As a warm-up in Part 1, the author gives an overview of calculus in the plane, with the intent of eventually defining the local degree of a mapping from an open set in the plane to another. This is done in the second part of the book, where winding numbers are defined, and the important concept of homotopy is introduced. These concepts are shown to give the fundamental theorem of algebra and invariance of dimension for open sets in the plane. The delightful Ham-Sandwich theorem is discussed along with a proof of the Lusternik-Schnirelman-Borsuk theorem. I would like to see a constructive proof of this theorem, but I do not know of one.

Part 3 is the tour de force of algebraic topology, for it covers the concepts of cohomology and homology. The author pursues a non-traditional approach to these ideas, since he introduces cohomology first, via the De Rham cohomology groups, and these are used to proved the Jordan curve theorem. Homology is then effectively introduced via chains, which is a much better approach than to hit the reader with a HOM functor.Part 4 discusses vector fields and the discussion reads more like a textbook in differential topology with the emphasis on critical points, Hessians, and vector fields on spheres. This leads naturally to a proof of the Euler characteristic.

The Mayer-Vietoris theory follows in Part 5, for homology first and then for cohomology.

The fundamental group finally makes its appearance in Part 6 and 7, and related to the first homology group and covering spaces. The author motivates nicely the Van Kampen theorem. A most interesting discussion is in part 8, which introduces Cech cohomology. The author's treatment is the best I have seen in the literature at this level. This is followed by an elementary overview of orientation using Cech cocycles.

All of the constructions done so far in the plane are generalized to surfaces in Part 9. Compact oriented surfaces are classified and the second de Rham cohomology is defined, which allows the proof of the full Mayer-Vietoris theorem.

The most important part of the book is Part 10, which deals with Riemann surfaces. The author's treatment here is more advanced than the rest of the book, but it is still a very readable discussion. Algebraic curves are introduced as well as a short discussion of elliptic and hyperelliptic curves.

The level of abstraction increases greatly in the last part of the book, where the results are extended to higher dimensions. Homological algebra and its ubiquitous diagram chasing are finally brought in, but the treatment is still at a very understandable level.

For examples of the author's pedagogical ability, I recommend his book Toric Varieties, and his masterpiece Intersection Theory.

This is one of the great algebraic topology books!
This is a book for people who want to think about topology, not just learna lot of fancy definitions and then mechanically compute things. Fulton hasput the essence of Algebraic Topology into this book, much in the way MikeArtin has done with his "Algebra". In my opinion, he should winsome sort of expository award for it.

Probably better as a 2nd (or 3rd) course rather than 1st
Most mathematicians, I suspect, can relate to the "colloquium experience": the first minutes of a lecture go easily, followed by twenty or thirty of real edification, concluded by ten to fifteen of feeling lost.I regret to say that this was pretty much my experience with the book.Fulton writes with unusual enthusiasm and the first two- thirds of the book is a joy to read, even while it is real work.I imagine that he must be a remarkable teacher in person.He has some threads such as winding numbers and the Mayer-Vietoris Sequence that continue throughout the book, bringing unity to a wide selection of topics.There are a number of applications of the subject to other areas, such as complex analysis (Riemann surfaces) and algebraic geometry (the Riemann-Roch Theorem), to name only two.There are particularly interesting illustrations of the Brouwer Fixed Point Theorem and related results.Unfortunately, there are two rather major reservations I have about the book.The first, already alluded to, is that it seemed to me to become precipitously difficult towards the end.The second is that this book would be excellent for a second or perhaps third course in the subject rather than a first.While the topics he covers are interesting in their own right, I still favor a more "standard" approach covering simplicial complexes, homology, CW complexes, and homotopy theory with higher homotopy groups, such as in the books by Maunder, Munkres, or Rotman (the last two of which I recommend unreservedly).It is true that Fulton has some coverage these topics, and a particularly extensive discussion of group actions and G-spaces, but he presupposes a background or ability that the novice to algebraic topology is unlikely to have.I would like to recommend this book, as I found it very edifying, but it seems better suited for one with some prior acquaintance to the subject. ... Read more

Product Description

 Learn the basics of point-set topology with the understanding of its real-world application to a variety of other subjects including science, economics, engineering, and other areas of mathematics.   Introduces topology as an important and fascinating mathematics discipline to retain the readers interest in the subject. Is written in an accessible way for readers to understand the usefulness and importance of the application of topology to other fields. Introduces topology concepts combined with their real-world application to subjects such DNA, heart stimulation, population modeling, cosmology, and computer graphics. Covers topics including knot theory, degree theory, dynamical systems and chaos, graph theory, metric spaces, connectedness, and compactness. A useful reference for readers wanting an intuitive introduction to topology.

Customer Reviews (5)

Very understandable
Everything is well explained with lots of pictures and examples. I am able to read the text and learn the material without having to resort to outside reference material.

Not for Math Majors
This book would be good for someone who has not had prior experience with proof-based mathematical courses before.It is full of examples, pictures and logically sound proofs.

Having said that, this book is NOT a sufficient introduction to topology for even a moderately advanced undergraduate in mathematics such as myself.It's proofs are wordy and convoluted which make them difficult to read and it inexplicably lacks many important proofs and theorems that are proved in other standard introductory textbooks (like for, example, the proof of the Urysohyn Metrization Theorem, proved in Munkres). At times the logic is so pedantic and wordy that I find it makes simple proofs difficult to read. Do yourself a favor and learn from one of the classics.

Fabulous Introduction to Topology!
I purchased this book upon recommendation from an internet buddy.I'm currently taking my first topology course (at an undergraduate level) and using Topology (2nd Edition) as the assigned text.I understand that Munkres is the "standard", and I don't have any real complaints about it, but I wanted something else to help broaden my understanding, and Adams and Franzosa did a great job in providing a book that does exactly that.

While Munkres presents everything from a very mathematically rigorous point of view, it took me several chapters before I really understood what we were talking about in a sense other than developing a branch of mathematics.It's great to follow theorems and definitions, but Munkres left me sort of mystified as to why we were doing this for quite some time.On the other hand, this book is all about the why and the how.

Applications of topology are presented from the get-go, usually as sections appended to the chapter that introduces the concept, so that the applications are more of an optional exploration than a focus.This really helps to motivate the reader and highlight the important concepts; it also makes it much easier to explain to a curious friend what exactly it is that you're doing.

Rigorous definitions and theorems are almost always accompanied by a plainer explanation of what exactly we're working with and why, and some of the diagrams, especially in the sections on quotient maps, are invaluable in visualizing what's going on and keeping track of what's a subset of what being mapped to where.This book does a good professor's job--instead of merely regurgitating theory and leaving you to put the pieces together, it's an excellent guide to a deeper understanding of the subject.

The exercises are plentiful and well-chosen.The authors gently guide the reader along the right path when asking for a new proof, and there are enough examples given to help the reader broaden her thinking to new approaches.The last several chapters go into more detail about specific topics that take the general concepts in advanced directions; this structure avoids breaking the flow of information.Overall, the text is very well-organized, and the authors have painstakingly highlighted suggested paths through the material.

This is, however, an introductory text, and it sticks mostly to point-set topology.There are a few results that I was surprised to see missing, and a few concepts that were skipped entirely (for instance, the distinction between the product topology and the box topology--the box topology is not discussed).For my purposes, it everything I could hope for--a patient discussion that expanded and clarified the topics I've already encountered using Munkres.As an introductory text, I couldn't imagine anything better.

An excellent text with supporting website
I was a student of Dr. Franzosa when this book was nothing more than a word document.I've been priveledged to watch it grow, draft by draft, into the complete text that it is now. As a student of topology I find this text refreshing, as the applications bring the theorems and lemmas and corollaries to life.Its written clearly, well illustrated and just plain fun. Another useful tool is a supporting website http://germain.umemat.maine.edu/faculty/franzosa/ITPA.htm

A great introduction to pure and applied topology
Although this book is a great introduction to pure and applied topology with several examples, figures and exercises making it is a good option for self-learning, I believe that the main differential of this book is the applied part of the book where one may find applications in economics, dynamical systems, graph theory etc. Furthermore, in the preface of the book, the author shows the minimal path that you have to follow in order to have the minimal necessary knowledge to understand the applied part of the book.

On the other hand, if you are interested only in pure topology, due to the difference of price I suggest you the Theodore W. Gamelin and Robert Everist Greene's Introduction to Topology, which is also a very nice book and very much cheaper.
... Read more

Product Description
Excellent text offers comprehensive coverage of elementary general topology as well as algebraic topology, specifically 2-manifolds, covering spaces and fundamental groups. The text is accessible to students at the advanced undergraduate or graduate level who are conversant with the basics of real analysis or advanced calculus. Problems, with selected solutions. Bibliography. 1975 edition.
... Read more

Customer Reviews (5)

Good book / Great supplement
I used this book as a supplement for math590(ind.study) on simplicial homology groups. This book is good because its cheap, concise, and rigorous (and characterizes all compact surfaces as well as proves the simplicial approximation theorem). There are hints and answers to the exercises which are numerous and vary in difficulty. I think this is the best all around book Dover offers on topology; Maunders is completely in a league of its own.

Who cares who writes the reviews
It is the author's ability to properly convey the ideas of the given subject that should be addressed not who the author chose to review it.Having read books by several of the "well-known' authors in the area of Topology, i.e. Munkres, Kelley and Bourbaki, I found this book quite informative, lively and lives up to the author's assertion that there is a definte need for books that are less dense(terse, pedantic) and which get right to the point, illustrating and presenting the material essential for an introductory exposure to Topology.The exercises are well chosen and extend the material presented in the text which is a complementary bonus since there appears an unfortunate trend in some texts to have seemingly irrelevant exercises at the end of each section.Also I found this to be good book for independent study and strongly recommend it to all highly motivated undergraduates.

Not Funny, Professor
While a good sense of humor is as necessary to reading (and hopefully to learning) topology as is mental acuity, self-serving Reviews by Authors are offensive and similar "contributions" by Authors' relatives (ifthat is in fact what we have here)are even more so given the price oftextbooks these days. Why doesn't the good professor submit the results ofa survey of his students (anonymous and conducted by someone other thanhimself, we would hope)who have used this book as a text? Better yet, wouldthe professor submit an open letter to a peer-reviewed journal solicitingsuch evaluations from colleagues at other schools which haveused his text?

Excellent Text--Dad you were right.
I find this to be an excellent treatment of concepts with have been made quite difficult in other texts.

Dad approves!
As the father of the author, I recommend this book heartily to topologists the world over.I originally purchasedmy copy at Blackwell's bookshop in Oxford, England. --Irving Kah ... Read more

Product Description
Superb introduction to rapidly expanding area of mathematical thought. Fundamentals of metric spaces, topologies, convergence, compactness, connectedness, homotopy theory and other essentials. Numerous exercises, plus section on paracompactness and complete regularity. References throughout. 107 illustrations.
... Read more

Customer Reviews (8)

Not really elementary but fine for self-study
I agree with the earlier quite positive reviews: this is a clearly written, concise introduction with plenty of examples. But let's be clear: this book is not really elementary.It's at the level of Munkres Topology (2nd Edition) and requires more mathematical maturity than, say, Mendelson Introduction to Topology: Third Edition, which by the way is an especially reader-friendly first book (but note Mendelson leaves out some key topics, e.g. separation axioms, metrization theorems and function spaces).

Like Mendelson, and unlike Munkres, Gemignani first introduces metric spaces, which are more familiar and intuitive than general topological spaces, because they underlie the usual mathematics in Euclidean n-space we are all familiar with, and then generalizes to general topological spaces. I think this concrete-to-abstract organization particularly lends itself to self-study since people naturally generalize from concrete or restricted cases to general cases. But my very positive evaluation of the book also has a lot to do with the fact that I found Chapter 6 on Convergence superb: it introduces, and provides concrete motivation for, a notion of convergence more general than one based on sequences (suitable for metric spaces) and then develops in some detail, two provably equivalent approaches: the theory of nets and the theory of filters (the latter of which one meets in lattice theory and model theory). (You can also find a discussion of convergence in general topologies in Ch 4. of Willard (1970) General Topology, another earlier book (1970) worth considering.) Those familiar with Isham's Modern Differential Geometry for Physicists (World Scientific Lecture Notes in Physics) might recall that he uses filters in his statement of convergence in general topologies (p. 29 and passim). But Isham's discussion is too abbreviated for one to really understand the material on first exposure: solution - read Chapter 6 Convergence in Gemignani and you're good to go! In contrast, Munkres does not even mention filters, and nets are relegated to Supplemental Exercises (pp. 187-188 in the 2nd edition). Generalized convergence, for some reason beyond me, is simply not treated by Munkres (if you know why or if I missed something, please let me know via a comment to this review.).

A few reviewers have complained about some out-dated notation but personally I hadn't even noticed even though I also have read Mendelson, Munkres and Willard. It is true though that in 1972, when Gemignani was published, terminology in topology had not, as he mentions, been completely standardized, e.g., he does not mention the term "boundary", but he defines "frontier" (p. 55), a term that does not appear in either Mendelson (1975) or Munkres (2nd ed., 1975). Also, there's a very nice section on Identification Spaces (79-85), which would now be called Quotient Spaces (cf. p. 80 and Munkres p. 137). These strike me as minor issues but if your easily frustrated by such differences in terminology or notation, then perhaps a newer book would be a better choice.

In summary: this book deserves careful consideration if you're in the market for a well-written, reasonably complete but concise general topology book with lots of examples that is also very inexpensive.My appreciation for Gemignani has only grown over the years as I have returned to it multiple times for a refresher.

classic introduction
I took a Topology class as an undergraduate and we used this text, the professor who specialized in topology told us on the first day of class that he hated every topology book that is on the market and he picked this book out as a reference because then "if it is a mistake, at least it is a cheap mistake."

Overall I think this book does a good job with introducing the concepts of basic topology, I have to say though that the notations are so archaic I found myself incorrectly trying to prove something that was false to begin with, because I read the symbols wrong.

Too Abbreviated
This book is only helpful for those who already have a solid foundation in Topology.It is a good quick-reference guide but the explanations are too short to provide any self-teaching capability.This book has no solutions to problems and most of them are ambiguous proofs with limited examples to help.Graphs and pictures are limited.I would expect books for math theory to have plenty of visualizations instead of expecting students to understand the overlapping and increasing jargon on every page.

A model of textbook writing
Gemignani's book explains everything thoroughly and clearly. What's best about it, though, is that the examples are chosen to be do-able and to lock in the material just gone over in the chapter.

Anyone who is a student of higher math and wishes to get a good start on topology will benefit greatly from working through this book, and they will enjoy it too.

A wonderful introduction
I used this book for my first course in topology.It's easy to read and well organized.While it is an introduction, the author covers more than other books in it's price range (Baum or Mendelson, for example).The author starts with some set theory basics and then moves to metric spaces, which he uses as motivation for his definition of a topology (Baum and some others use neighborhood systems instead).He then discusses homeomorphism, seperation axioms (T0 through T4), compactness and other topics.The exercises provide a nice level of challenge; they require thought but aren't impossible.I'd recommend this book as a reference, or to anyone interested in higher mathematics.And since it's a Dover book, the prices is right! ... Read more

Product Description
Applications from condensed matter physics, statistical mechanics and elementary particle theory appear in the book. An obvious omission here is general relativity--we apologize for this. We originally intended to discuss general relativity. However, both the need to keep the size of the book within the reasonable limits and the fact that accounts of the topology and geometry of relativity are already available, for example, in The Large Scale Structure of Space-Time by S. Hawking and G. Ellis, made us reluctantly decide to omit this topic. ... Read more

Customer Reviews (6)

Excellent overview and graphical explanation
This book shows you the geometric view of some advanced mathematical topics. It can greatly assist your intuition of what is going on in a mathematical setting when reading a true mathematics book. Armed with this book the other advanced text in Topology, Algebraic Geometry and Differential Geometry make more sense from a Physics point of view.

Excellent overview and graphical explanation
This book shows you the geometric view of some advanced mathematical topics. It can greatly assist your intuition of what is going on in a mathematical setting when reading a true mathematics book. Armed with this book the other advanced text in Topology, Algebraic Geometry and Differential Geometry make more sense from a Physics point of view.

Good attempt
When reading this book one can both admire these authors and feel sympathy with them. They have made an honest effort to explain the conceptsof differential geometry and topology in a way that is understandable and appreciated by the physicist reader. But the book falls short in many places, although there are some places where they do a fine job. They have taken on a very difficult project in this book, for it is quite straightforward to expound on the formalism of mathematics, but explaining it in a way that grants insight into its conceptual meaning is another matter altogether. Many physicists complain, with justification, that the way mathematics is presented in textbooks is not sufficient for giving them a deep appreciation of the underlying ideas involved. This, they argue, is what is needed for devising new physical theories and results based on these ideas. Physicists must assimilate very complex mathematical ideas very quickly in order to formulate these theories in a reasonable time frame. This is especially true in high energy physics, which in the last two decades has used mathematics like it has never been used before. Indeed, the mathematical complexity of high energy physics is dizzying, and if progress is going to be made in this field by the students of the 21st century, they are going to need mathematics books and documents that are more than just formal expositions. But, again, writing these kinds of books is very hard to do, and has yet to be done in a book to this date, although there are helpful discussions scattered throughout the mathematical literature.

Some of the concepts that need more in-depth explanation include: the theory of characteristic classes, sheaf theory, the theory of schemes in algebraic geometry, and spectral sequences in algebraic topology. There are of course many others, and some of the ones that the authors do a fairly good job of explaining in this book include: 1. the reason that the continuity of a function is defined in terms of inverses of open sets; 2. The orientability of a manifold; 3. The fundamental group and its relation with the first homology group. 4. The discussion on Morse theory.

Covers a lot of ground . . . but not always well
Unlike many physics students, I grant a lot of leeway to books on mathematics for physicists.I think it's all right for an author to engage in hand-waving arguments if this enhances physical intuition or even to make the occasional statements without proof if this allows more ground to be covered.However, if a proof actually is presented, I expect this proof to be correct.In this book, proofs are sometimes only for special cases of theorems stated more generally and often contain logical errors.

flawed and incomplete
Nash's book commits the sin many mathematical physics textbooks out there commit: "oh, we're writing for dimwit physicists, lets just give them a few scrawny examples and assure them everything else works alright." I'm sorry but writing for physicists is NOT an excuse for writing a sloppy textbook. Would you feel alright not knowing how an integral is defined? Would you use a numerical evaluation software to calculate integrals in serious research without understanding the algorithm it uses? If you do then you're a pretty shoddy physicist. I'm not saying this out of some "macho" sentiment many purist physicists have - I'm simply saying this because I feel the way this book teaches you diff. geometry is wrong - it teaches you to draw pictures and go by the pictures. When the pictures run out, so does your understanding.

This book is supposed to teach differential geometry. However, very little can be learned from it unless one already knows differential geometry: definitions are sometimes not general and sometimes not present at all, theorems are often stated only for special cases and even more often than that not proved at all. Sure, the book offers nice geometrical intuition, but this is not enough. An example: the book "proves" Stoke's theorem around page 40. Now, even a rigorous and condensed book would have problems doing that, considering the amount of "machinery" one needs to build up for it (tensors, differential forms, manifolds and so forth). This means the book makes a mess of it - big time.
There are many fine diff. geometry books out there, some for physicists, some not, which you should check out - Nakahara's text is so much better. For geometrical intuition I suggest picking up Schutz's book. Several books from the GTM (Graduate texts in mathematics series, the yellow ones) are really very accessible, such as Introduction to Topological Manifolds/Smooth Manifolds. Another good one is Allen Hatcher's Algebraic Topology for homotopy, homology and cohomology. For a good and responsible exposition, do yourself a favor and look for something else. ... Read more

Product Description

From the reviews of the 1st edition:

"This book provides a comprehensive and detailed account of different topics in algorithmic 3-dimensional topology, culminating with the recognition procedure for Haken manifolds and including the up-to-date results in computer enumeration of 3-manifolds. Originating from lecture notes of various courses given by the author over a decade, the book is intended to combine the pedagogical approach of a graduate textbook (without exercises) with the completeness and reliability of a research monograph…

All the material, with few exceptions, is presented from the peculiar point of view of special polyhedra and special spines of 3-manifolds. This choice contributes to keep the level of the exposition really elementary.

In conclusion, the reviewer subscribes to the quotation from the back cover: "the book fills a gap in the existing literature and will become a standard reference for algorithmic 3-dimensional topology both for graduate students and researchers".

Zentralblatt für Mathematik 2004

For this 2^nd edition, new results, new proofs, and commentaries for a better orientation of the reader have been added. In particular, in Chapter 7 several new sections concerning applications of the computer program "3-Manifold Recognizer" have been included.

Product Description

From reviews of the first edition:

"In the world of mathematics, the 1980's might well be described as the "decade of the fractal". Starting with Benoit Mandelbrot's remarkable text The Fractal Geometry of Nature, there has been a deluge of books, articles and television programmes about the beautiful mathematical objects, drawn by computers using recursive or iterative algorithms, which Mandelbrot christened fractals. Gerald Edgar's book is a significant addition to this deluge. Based on a course given to talented high- school students at Ohio University in 1988, it is, in fact, an advanced undergraduate textbook about the mathematics of fractal geometry, treating such topics as metric spaces, measure theory, dimension theory, and even some algebraic topology...the book also contains many good illustrations of fractals (including 16 color plates)."

Mathematics Teaching

"The book can be recommended to students who seriously want to know about the mathematical foundation of fractals, and to lecturers who want to illustrate a standard course in metric topology by interesting examples."

Christoph Bandt, Mathematical Reviews

"...not only intended to fit mathematics students who wish to learn fractal geometry from its beginning but also students in computer science who are interested in the subject. Especially, for the last students the author gives the required topics from metric topology and measure theory on an elementary level. The book is written in a very clear style and contains a lot of exercises which should be worked out."

H.Haase, Zentralblatt

About the second edition: Changes throughout the text, taking into account developments in the subject matter since 1990; Major changes in chapter 6. Since 1990 it has become clear that there are two notions of dimension that play complementary roles, so the emphasis on Hausdorff dimension will be replaced by the two: Hausdorff dimension and packing dimension. 6.1 will remain, but a new section on packing dimension will follow it, then the old sections 6.2--6.4 will be re-written to show both types of dimension; Substantial change in chapter 7: new examples along with recent developments; Sections rewritten to be made clearer and more focused.

Customer Reviews (4)

some new material->same approach
I bought the first edition of this in the early 90's
and was disappointed that it didn't have the Mandelbrot or other complex dynamics
in it. Dr. Edgar has updated the older book with Julias, multifractals
and Superfractals, but has stayed true to his topological measure theory
Hausdorff space approach. He never updates his Biscovitch-Ursell functions
to 2d and 3d parametrics or the unit Mandelbrot cartoon method.
Some of his definitions are still so minimal
that duplicating the fractals needs much more information?!
The text is still the good place to begin, but
it is a shame that Dr. Edgar has not kept up
with many of the developments in the field. Zipf and Per Bak
are left out, but my double V L-system made the index as a picture.

A nice book
I liked this book because it provided me with a new perspective on metricspaces, in using them as a basis learning about fractals. I think it servesas a nice book for an undergraduate to read and get enthused about studyingfractals at a higher level.

Good starting point to study fractal geometry.
This book could be used as a bridge between traditional books on topology-analysis and the speciallized treatises on fractal geometry. More a catalog of definitions, methods, and references than a course text, itcovers the fundamental topological and measure-theoretic concepts needed tounderstand the principles of some of the different dimension theories thatexist. But warning: the book is far away of being a complete exposition onany of the subjects it includes.

Suitable for 3rd-year undergrads.Interesting examples and exercises. Extensive bibliography.

Please checkmy other reviews in my member page (just click on my name above).

A difficult but worthy book!
The programs are in LOGO: don't let the turtles fool you, this is the real stuff by a master teacher. It is hard and the examples are even harder. The problem sets are at times impossible, but in the end Dr. Edgar delivers: understanding! Your unique Associates ID is:thefractaltransl. ... Read more

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Customer Reviews (1)

Excellent complement for General Topology
I discovered this book as a Cambridge Maths undergraduate when it first came out in the early 1970s and it really helped complement the recommended texts and lecture notes. It is full of well explained examples, graded exercises and clear exposition and proofs. An especially interesting feature is the set of 42 tables of theorems and counterexamples in the appendix, doubtless slightly out of date now but still invaluable to those who, in the words of the author, absolutely ~must~ know whether every locally compact, completely regular, separable space is sigma-compact. (Although, to be totally honest, I have still never managed to find the answer to that particular combination, I don't consider this justifies reducing the overall 5-star rating).

If I had not still got my 1972 copy, I would definitely buy this Dover reprint, even if it is slightly more expensive now than the £6.30 I paid for the hardback original. Be warned though, this is topology for analysis, as it says on the tin. The intuitions are almost exclusively analytically motivated and those in search of discussions of doughnut geometry or notions (co)homological and/or homotopic will need to search elsewhere. ... Read more

Product Description
The single most difficult thing one faces when one begins to learn a new branch of mathematics is to get a feel for the mathematical sense of the subject. The purpose of this book is to help the aspiring reader acquire this essential common sense about algebraic topology in a short period of time. To this end, Sato leads the reader through simple but meaningful examples in concrete terms. Moreover, results are not discussed in their greatest possible generality, but in terms of the simplest and most essential cases.

In response to suggestions from readers of the original edition of this book, Sato has added an appendix of useful definitions and results on sets, general topology, groups and such. He has also provided references.

Topics covered include fundamental notions such as homeomorphisms, homotopy equivalence, fundamental groups and higher homotopy groups, homology and cohomology, fiber bundles, spectral sequences and characteristic classes. Objects and examples considered in the text include the torus, the Möbius strip, the Klein bottle, closed surfaces, cell complexes and vector bundles. ... Read more

Customer Reviews (5)

"Intuition" more a prerequisite than a result
This was my first crack at algebraic topology, self-studying long after my university days. I thought I'd read this book as a warm-up for Bott & Tu. The book is written in the laid-back discursive style that is one of the more charming attributes of Japanese math books. It's also short, and the author has provided solutions or hints for most of the modest exercises. At first glance it looks like a pleasant way to spend a few afternoons in a cafe.

But appearances can be deceiving. The intuitions referred to are not those of a typical beginner. No less disingenuous is the occasional advice saying it's OK to skip a chapter: the concepts and definitions are inevitably used in later ones. These are what Japanese call "tatemae" -- the stuff that's said just for the sake of making a good (or at least better) impression.

The reviewers who suggested that the book supplements more advanced texts are closer to the mark. I found myself resorting to Bott & Tu and Hatcher to clear up concepts presented in this one, when I'd expected the reverse. E.g., Sato's explanation of exact sequences was ultra-concise and rather puzzling, while the two books I mentioned and even Wikipedia are quite helpful about them. B&T also uses many more diagrams when it counts, including in some clear and beautiful proofs about homotopies that Sato presents in a drier style. Nor does Sato do a good job of motivating why cohomology is more useful than homology; for all its shortcomings (including lack of coverage of De Rham cohomology), even 1970's-vintage Maunder does a better job at this. (The first few pages of Hatcher's Chapter 3 are even better on that point, but that's what one would expect from such a humongous book.) And the diagrams accompanying the description of fiber bundles don't even indicate a fiber; there are many more "intuitive" explanations of this topic elsewhere.

This may be a good tool for reinforcing material you have learnt or are learning from another source. But you might not find it as suitable for a free-standing introduction as the title and a casual inspection might suggest. I give it 3.5-4 stars instead of 3 as a handicap, considering my own amateurism, and also because of the good range of topics touched on.

Excelent Start
In my opinion, this is a great little book to take with you to a park or on a trip to read before you start tackling a more serious book such as the one by Allen Hatcher.This book will give you a great over view of many major topics in Algebraic Topology; for a serious reader, you might want to read this book in parallel with Hatcher, Massey and Munkres (Topology, 2nd Edition).I find that these three books compliment one another very well if you are trying to learn this beautiful subject on your own.I use Sato's book to read about general ideas; once I understand the surface of the concepts I then reference the latter two books to dive deeper into the machinery.It's working well for me; however, do not be fooled, nothing replaces a great teacher!

Excellent accompaniment to Hatcher
As a student just wading into the realm of Algebraic Topology, this book has been a wonderful companion. If you are looking for a book that will lay out precise proofs of theorems and get down to the nity-gritty, this book is not for you. However, if you are new to A.T. as I am, and want a book that will give you a nice easy to follow introduction to a topic before wading into your thicker text, then this book will help you tremendously. For instance, reading the chapters regarding CW-complexes and Homotopy in Sato, although thin and easy to follow (you will have to do a little bit of lifting, but not too much), helped me to more easily digest what was to come in Chapters 0 and 1 of Hatcher (which I also highly recommend, incidentally). It always helps to read material taken from a different person's perspective, and Sato has truly made Algebraic topology more transparent in this brief overview.

Good Supplementary Reading
This modest 118-page book would best accompany one of the standard graduate texts -- Spanier, Dold, Switzer, Massey, Husemoller,Maunder, Munkres, Bott and Tu, Bredon, or Greenberg and Harper. It can't be used as a text.

The book presents the most basic ideas pertaining to homotopy, homology, cohomology, fibre bundles, spectral sequences, and characteristic classes. The emphasis is on simple examples and simple calculations to demonstrate what is going on. Rigorous definitions, proofs, and even frequently even the statements of theorems, are avoided.

One good aspect of the treatment is the axiomatic presentation of homology and cohomology a la Eilenberg and Steenrod. Some of the essential material is also presented, e.g. the cup product that gives a ring structure to the cohomology group, the Kunneth theorem, the Universal Coefficient theorem, and so on.

The book would afford a bird's-eye view, a conspectus, to a bright undergraduate or beginning graduate student. It goes without saying, of course, that this is for motivation, and it doesn't replace the hard technical grind required to master the subject.

The book suffers in comparison to the one by Fomenko, Fuchs and Gutenmacher (Homotopic Topology), but that, alas, can't be had for love or money.

algeblaic topology
there are much examples. so good to understand. ... Read more

Product Description
This self-contained introduction to algebraic topology is suitable for a number of topology courses. It consists of about one quarter 'general topology' (without its usual pathologies) and three quarters 'algebraic topology' (centred around the fundamental group, a readily grasped topic which gives a good idea of what algebraic topology is). The book has emerged from courses given at the University of Newcastle-upon-Tyne to senior undergraduates and beginning postgraduates. It has been written at a level which will enable the reader to use it for self-study as well as a course book. The approach is leisurely and a geometric flavour is evident throughout. The many illustrations and over 350 exercises will prove invaluable as a teaching aid. This account will be welcomed by advanced students of pure mathematics at colleges and universities. ... Read more

Customer Reviews (1)

Very well written introduction.
The book is a very well written introduction to algebraic topology. Itpresents the main basic results, provides geometric insights andcomputational tools. After reading it one really feels confident and onewhishes to go further : the book really is what the title says. It istherefore perfectly suited for a first reading. ... Read more