Customer Reviews (1)

Not for the Undergraduate
Professor Cullen says in the preface that this book should be read by the advanced undergraduate or first-year graduate student, but this book in my opinion should only be read by a first- or second-year graduate student. The mathematics in this book is as rigourous as math can possibly get; the proofs are often quite long and sometimes difficult, and the concepts Professor Cullen tries to convey are sometimes very difficult to follow. If you like your math rigourous (trust me, there are people out there that like rigourous mathematics) and you have some backround in topology and real analysis, then this might be for you. But remember, this book is serious mathematics, and if you try to pick this book up with no backround then you'll get eaten alive. ... Read more

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Customer Reviews (3)

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

Book Description
This book is designed for the reader who wants to get a general view of the terminology of General Topology with minimal time and effort. The reader, whom we assume to have only a rudimentary knowledge of set theory, algebra and analysis, will be able to find what they want if they will properly use the index. However, this book contains very few proofs and the reader who wants to study more systematically will find sufficiently many references in the book.

Key features:

• More terms from General Topology than any other book ever published

• Short and informative articles

• Authors include the majority of top researchers in the field

• Extensive indexing of terms ... Read more

Customer Reviews (2)

Good book, but overprized.
This book indeed covers the standar subject of topology, and is well written, self contained and elegant. For a topology class there is no much more for the teacher to do than to clearify the definitions, so the best advice is to get the book in order to be prepared for class.

The only problem with this book is the prize. I don't understand why publishing companies think that we are going to pay 120 bucks for a paperback edition of a book on a subject that is not even "state of the art - frontiers of the science" kind.

Be At One With The Topological Cosmos
I used this textbook to supplement Dr. Cain's notes in a topology class at Georgia Tech.I found it to be an excellent introduction to the subject.Theorems are proved with clarity, and the exercises inspire thought about the subject matter.This book was the first place I encountered a proof of the Tychonoff Product Theorem. The proof was presented very well, in my opinion.In addition, there is no shortage of examples in each section.I would heartily recommend this book to anyone with an interest in learning general topology. ... Read more

Customer Reviews (2)

Written for the physicist in mind
This book, written by some of the master expositors of modern mathematics, is an introduction to modern differential geometry with emphasis on concrete examples and concepts, and it is also targeted to a physics audience. Each topic is motivated with examples that help the reader appreciate the essentials of the subject, but rigor is not sacrificed in the book.

In the first chapter the reader gets a taste of differentiable manifolds and Lie groups, the later gving rise to a discussion of Lie algebras by considering, as usual, the tangent space at the identity of the Lie group. Projective space is shown to be a manifold and the transition functions explicitly written down. The authors give a neat example of a Lie group that is not a matrix group. A rather quick introduction to complex manifolds and Riemann surfaces is given, perhaps too quick for the reader requiring more details. Homogeneous and symmetric spaces are also discussed, and the authors plunge right into the theory of vector bundles on manifolds. Thus there is a lot packed into this chapter, and the authors should have considered spreading out the discussion more, as it leaves the reader wanting for more detail.

The authors consider more fundamental questions in smooth manifolds in chapter 3, with partitions of unity used to prove the existence of Riemannian metrics and connections on manifolds. They also prove Stokes formula, and prove the existence of a smooth embedding of any compact manifold into Euclidean space of dimension 2n + 1. Properties of smooth maps, such as the ability to approximate a continuous mapping by a smooth mapping, are also discussed. A proof of Sard's theorem is given, thus enabling the study of singularities of a mapping. The reader does get a taste of Morse theory here also, along with transversality, and thus a look at some elementary notions of differential topology. An interesting discussion is given on how to obtain Morse functions on smooth manifolds by using focal points.

Notions of homotopy are introduced in chapter 3, along with more concepts from differential topology, such as the degree of a map. A very interesting discussion is given on the relation between the Whitney number of a plane closed curve and the degree of the Gauss map. This leads to a proof of the important Gauss-Bonnet theorem. Degree theory is also applied to vector fields and then to an application for differential equations, namely the Poincare-Bendixson theorem. The index theory of vector fields is also shown to lead to the Hopf result on the Euler characteristic of a closed orientable surface and to the Brouwer fixed-point theorem.

Chapter 4 considers the orientability of manifolds, with the authors showing how orientation can be transported along a path, thus giving a non-traditional characterization as to when a connected manifold is orientable, namely if this transport around any closed path preserves the orientation class. More homotopy theory, via the fundamental group, is also discussed, with a few examples being computed and the connection of the fundamental group with orientability. It is shown that the fundamental group of a non-orientable manifold is homomorphic onto the cyclic group of order 2. Fiber bundles with discrete fiber, also known as covering spaces, are also discussed, along with their connections to the theory of Riemann surfaces via branched coverings. The authors show the utility of covering maps in the calculation of the fundamental group, and use this connection to introduce homology groups. A very detailed discussion of the action of the discrete group on the Lobachevskian plane is given.

Absolute and relative homotopy groups are introduced in chapter 5,and many examples are given of their calculation. The idea of a covering homotopy leads to a discussion of fiber spaces. The most interesting discussion in this chapter is the one on Whitehead multiplication, as this is usually not covered in introductory books such as this one, and since it has become important in physics applications. The authors do take a stab at the problem of computing homotopy groups of spheres, and the discussion is a bit unorthodox since it depends on using framed normal bundles.

The theory of smooth fiber bundles is considered in the next chapter. The physicist reader should pay close attention to this chapter is it gives many insights into the homotopy theory of fiber bundles that cannot be found in the usual books on the subject. The discussion of the classification theory of fiber bundles is very dense but worth the time reading. Interestingly, the authors include a discussion of the Picard-Lefschetz formula, as an example of a class of "fiber bundles with singularities". Those interested in the geometry of gauge field theories will appreciate the discussion on the differential geometry of fiber bundles.

Dynamical systems are introduced in chapter 7, first as defined over manifolds, and then in the context of symplectic manifolds via Hamaltonian mechanics. Liouville's theorem is proven, and a few examples are given from relativistic point mechanics. The theory of foliations is also discussed, although the discussion is too brief to be of much use. The authors also consider variational problems, and given its importance in physics, they continue the treatment in the last chapter of the book, giving several examples in general relativity, and in gauge theory via a consideration of the vacuum solutions of the Yang-Mills equation. The physicist reader will appreciate this discussion of the classical theory of gauge fields, as it is good preparation for further reading on instantons and the eventual quantization of gauge fields.

A masterful sequel!
Novikov et al's first volume was the defining book on differential geometry (S-V 93). The second volume picks up on the detailed theory of manifolds and topology and other advanced theories of differentialgeometry, including homotopy groups, Lie algebras and digressing intophysical theories as well (eg.Yang-Mills) giving one of the juciest bookson the subject - an utter delight! ... Read more

Book Description

Customer Reviews (1)

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

... Read more

Book Description
The goal of this publication is to provide basic tools of differential topology to study systems of nonlinear equations, and to apply them to the analysis of general equilibrium models with complete and incomplete markets. The main content of general equilibrium analysis is to study existence, (local) uniqueness and efficiency of equilibria. To study existence Differential Topology and General Equilibrium with Complete and Incomplete Markets combines two features. As a first step, first order conditions (of agents' maximization problems) and market clearing conditions, instead of aggregate excess demand functions, are used. As a second step, a homotopy argument, stated and proved in relatively elementary manner, is applied to that "extended systemof equations. Local uniqueness and smooth dependence of the endogenous variables from the exogenous ones are studied using a version of a so called parametric transversality theorem. In a standard general equilibrium model, all equilibria are efficient, but that is not the case if some imperfection, like incomplete markets, asymmetric information, strategic interaction, is added. Then, for almost all economies, equilibria are inefficient, and an outside institution can Pareto improve upon the market outcome. Those results are proved showing that a well-chosen system of equations has no solutions.
The target audience of Differential Topology and General Equilibrium with Complete and Incomplete Markets consists of researchers interested in economic theory. The needed background is multivariate analysis, basic linear algebra and basic general topology. ... Read more

Book Description
This book with its three contributions by Arhangel'skii and Choban treats important topics in general topology and their role in functional analysis and axiomatic set theory. It discusses, for instance, the continuum hypothesis, Martin's axiom; the theorems of Gel'fand-Kolmogorov, Banach-Stone, Hewitt and Nagata; the principles of comparison of the Luzin and Novikov indices.The book is written for graduate students and researchers working in topology, functional analysis, set theory and probability theory. It will serve as a reference and also as a guide to recent research results. ... Read more

Book Description
Sheldon Davis' text is written for introductory courses in topology taken by advanced undergraduate and beginning graduate students.Designed to be flexible, the text is divided into two parts to accomodate different courses, course configurations, and instructor preferences.Part I of the text covers the bare essentials every student should know about topology before continuing on to study point-set or set-theoretic topology, algebraic topology, funcitonal analysis, continuum theory, or the many other important areas in mathematics that utilize topology fundamentals.To keep the text manageable for beginning students, use of set theory in Part I is kept to an intuitive level.Part II contains a complete beginning course in general topology, or set-theoretic topology.General topology courses that assume prior background in the fundamentals can start directly with Part II and use the material in Part I for conceptual review.

This text is part of the Walter Rudin Student Series in Advanced Mathematics. ... Read more

Book Description
This volume mainly focuses on various comprehensive topologicaltheories, with the exception of a paper on combinatorial topologyversus point-set topology by I.M. James and a paper on the history ofthe normal Moore space problem by P. Nyikos.
The history of the following theories is given: pointfree topology,locale and frame theory (P. Johnstone), non-symmetric distances intopology (H.-P. Künzi), categorical topology and topologicalconstructs (E. Lowen-Colebunders and B. Lowen), topological groups (M.G. Tkacenko) and finally shape theory (S. Mardesic andJ. Segal).
Together with the first two volumes, this work focuses on the historyof topology, in all its aspects. It is unique and presents importantviews and insights into the problems and development of topologicaltheories and applications of topological concepts, and into the lifeand work of topologists. As such, it will encourage not only furtherstudy in the history of the subject, but also further mathematicalresearch in the field. It is an invaluable tool for topologyresearchers and topology teachers throughout the mathematical world. ... Read more

Product Description
This manuscript is a detailed presentation of the ten lectures given by the author at the NSF Regional Conference on Three-Manifold Topology, held October 1977, at Virginia Polytechnic Institute and State University. The purpose of the conference was to present the current state of affairs in three-manifold topology and to integrate the classical results with the many recent advances and new directions. ... Read more

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

Customer Reviews (10)

Great service!
The service overall was very good:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Topology Classic
This book was recommended for our analysis course (final year at Adelaide University). It helped me pass the course but more importantly, gave me an interest in metric spaces and topology. The book is an excellent communicator and nearly 20 years after I have read it I am looking out for a secondhand copy! ... Read more