Book Description

This introduction to topology provides separate,in-depth coverage of both general topology and algebraic topology.Includes many examples and figures. GENERALTOPOLOGY. Set Theory and Logic. Topological Spaces and ContinuousFunctions. Connectedness and Compactness.Countability and SeparationAxioms. The Tychonoff Theorem. Metrization Theorems andparacompactness. Complete Metric Spaces and Function Spaces. BaireSpaces and Dimension Theory. ALGEBRAIC TOPOLOGY. The FundamentalGroup. Separation Theorems. The Seifert-van Kampen Theorem.Classification of Surfaces. Classification of Covering Spaces.Applications to Group Theory. For anyone needing abasic, thorough, introduction to general and algebraic topology andits applications.

Customer Reviews (34)

,,,
This is the standard into to point-set topology for a reason.It covers general topology very well, with easy-to-follow proofs and exercises that are actually possible to do.The problem is that the algebraic topology portion of this text (around 1/3 of the whole thing), is vastly inferior to Hatcher's "Algebraic Topology" book, which happens to be free.If you're looking for a good way to begin studying topology, then this is the book you're looking for, but if you want to learn about the fundamental group (and various related topics), then Allen Hatcher's book is available for free on his website.It calls to you...[insert spooky noises here]

Ideal
I've gone through most of this book and did many of the problems. The sections I skipped are the section on nets, the "review section" in chapter 4, the existence of a continuous nowhere-differentiable function, Dimension Theory, and all of Chapter 10 (Separation Theorems in the Plane), the Classification Theorem, and Constructing Compact Surfaces.

This is definitely my favorite math book. The two other books I've read this semester (Conway's Complex Analysis, and Rudin's Real and Complex Analysis) simply don't compare. In fact I'm afraid I'll always find fault with every other math book, after reading this one. There's alot of good expository prose, many examples and diagrams, and if you pay attention to details, and struggle to supply missing ones, you won't miss a beat and will succeed (unlike sometimes in Rudin's text). The problems are appropriate; very few are mindless, most do require a little thought, but a motivated student could solve most or all of them in a reasonable amount of time. There are no sudden breaks in proofs or in the text that are relegated as exercises (unless it's a repeat of a previous proof), and although results from previous exercises are sometimes used, he always states the necessary hypotheses. The book is self-contained - he begins with 70+ pages of naive set theory, for instance (not a prerequisite for the rest of the book).

I feel that reading this book and working its problems has given me a solid and comprehensive grounding in basic topology, and this book does go beyond what's usually taught in a first topology course, and the second half of the book is all algebraic topology. Here I found the review of abelian groups, free products and free groups to be extremely helpful, though I did still have to contemplate these alot on my own afterwards. The Seifert-Van Kampen theorem was also well-presented; he presents it as a pushout diagram. In the last chapter, as a nice application, he proves using linear graphs that subgroups of free groups are free.

I just simply love this book, but to be fair, I do have some minor qualms.

(1) There are a few obvious typos, and I didn't find more than six
(2) I believe one step in the proof of Lemma 68.9 is incorrect; this arises from a definitional issue of the subgroup generated by a subset. earlier, he assumed the subset was itself a subgroup, but now he's assuming it's arbitrary. the correct definition is on the next page, and the method of proof, with this definition, does give the right result; almost nothing changes in the proof
(3) In Theorem 68.4, the monomorphism and generating assumptions aren't necessary
(4) Problem #2 on page 438: I think the X_i should be path-connected, and Wikipedia is in agreement with this. I tried passing to path-components, which solved one problem but gave me others. On the other hand, if you assume path-connectedness, the proof is is the right level of difficulty.
(5) He gives an exercise regarding absolute retracts and adjunction spaces. I think he should've elaborated more on adjunction spaces, as it does involve new notions (e.g. free/topological union). Also his definition of adjunction space is incomplete, as compared to other definitions I found
(6) The book binding is horrible (it's the same with his other book, "Analysis on Manifolds"). If you're paying 100+ dollars for a book, you should expect to receive something very pretty, but the typesetting of this book is quite dull, and the book falls apart easily (mine is in many pieces).

In conclusion I highly recommend this book for self-study, and for seeing how math books can and should be written. I hope Munkres writes more textbooks, I'd read every single one of them.

Good for auto-didacts
I used this book to teach myself some topology. Not being a mathematician, I cannot really assess how it trades off rigour with accessibility, but I can recommend it for self-study. It starts more or less from zero, is pretty clear and provides some welcome intuition to supplement the proofs. The best thing about it is the large number of challenging exercises, solutions to which are readily available on the web ([...]).

First half is great, don't bother with the second half
There are 14 chapters in the book, but it is only known for the first 8.The first 8 chapters cover pretty much everything you ever wanted to, or will ever need to know for point-set topology.It's easy to read, makes sense, lots of examples, proofs, and doable exercises.Extremely thorough.Munkres likes to talk, and some of his informal language is pretty funny in some places.Go somewhere else for Algebraic Topology.Some basics on homotopy theory are here, but nothing at all on (co)homology.My main complaint about this book is that it fails to make topology as exciting as it really is.

The best approach to point-set topology and an excellent graduate text
This is my *absolute* favourite maths text of all time. In the years I've owned it, I have recommended this book to nearly every single individual in mathematics I know. The approach of the book is unlike other books on point-set topology. Most books begin with metric spaces, and build up the motivation towards general topology. However, this book drops you right into the deep end, and begins with the general topology. The main philosophy seems to be not about "metric" but about "metrisable." Munkres covers everything you would possibly want from a book on general topology. In part 2, he even delves a little into Algebraic topology. It has been years since I first opened my copy of Munkres, and I keep referring again and again! ... Read more

Book Description
With more than 30 million copies sold, Schaum's are the most popular study guide on the planet. Mathematics students around the world turn to this clear and complete guide to general topology for its through introduction to the subject, includingeasy-to-follow explanations of topology of the line and plane and topological spaces. With 650 fully solved problems and hundreds more to solve on your own (with answers supplied), this guide can help you spend less time studying while you make better grades! ... Read more

Customer Reviews (4)

Very good
The book is wonderful, recommends to all. Very good

Geraldo Tavares -Campinas/SP- Brasil.

OK as a supliment
If you're taking a basic topology class (or any other abstract math class), it is usefull to have other books to reference.This book was only OK as a supliment.I found that you had to be careful the definitions, as some topologists will axiomize ideas differently (I found this happened the most when dealing with seperation axioms).I found that there are a few dover books out there that are a little more readable (Gemignani's, Mendelson's and Willard's books all come to my mind).This book won't hurt, but there are other options within it's price range.

Great resource for college-level math students...
I wish I had known about this summary when I was taking my Fundamental Concepts in Math course at the university.I would have done a lot better!I agree with the previous reviewer about adding REA's Problem Solver book for a complete set.

The topics covered include: (1) Sets and relations, (2) Functions, (3) Cardinality and Order, (4) Topology of the line and plane, (5) Topological Spaces, (6) Bases and Subbases [sic], (7) Continuity and topological equivalence, (7) Metric and normed spaces, (8)Countability, (10) Separation axioms, (11) Compactness, (12) Product spaces, (13) Connectedness, (14) Complete metric spaces, and (15) Function spaces.There is an appendix about Properties of real numbers.

This is a technical book and not for a casual reader.I believe this will be difficult for a teach-yourself-to-know reader, unless the reader has considerable math experience.

Another book that I found helpful in Fundamental Concepts (not simply Topology) was Courant et al., What is mathematics? 2nd ed.

My Fundamental Concepts professor taught the course with only lecture notes, and I was completely lost.Had I had this outline, I think I would have done better, because this Outline has a number of solved problems and exercises.

Good luck.

If you are a good researcher, this is a good resource!
If you are taking an introductory Topology course, I recommend using this book. It is a little old, and some of the problems are not solved, but it will offer you an advantage when using your class text. You should use Lipschutz's book together with REA's Topology Problem Solver to give you full study advantage when taking on this most difficult subject. With some moderate study, you should do better on your assignements! ... Read more

Book Description
From the reviews:
"The author has attempted an ambitious and most commendable project. He assumes only a modest knowledge of algebraic topology on the part of the reader to start with, and he leads the reader systematically to the point at which he can begin to tackle problems in the current areas of research centered around generalized homology theories and their applications. ... The author has sought to make his treatment complete and he has succeeded. The book contains much material that has not previously appeared in this format. The writing is clean and clear and the exposition is well motivated. ... This book is, all in all, a very admirable work and a valuable addition to the literature...
(S.Y. Husseini in Mathematical Reviews, 1976) ... Read more

Customer Reviews (1)

This might take a while...
The earlier chapters are quite good; however, some ofthe advanced topics in this book are better approached (appreciated) after one has learned about them elsewhere,at a more leisurely pace.For instance, this isn't the best place to first read about characteristic classes and topological K theory (I would recommend, without much hesitation, the books by Atiyah and Milnor & Stasheff, instead).Much to my disappointment, the chapter on spectral sequences is quite convoluted. Parts of 'user's guide' by Mclearywould certainlycome in handy here (which sets the stage rather nicely for applications).

So it turns out that supplemental reading (exluding Whitehead's massive treatise) is necessary to achieve a better understanding of algebraic topology at the level of this book.The homotopical view therein will be matched (possibly superseded) by Aguilar's book (forthcoming, to which I am very much looking forward).

Good luck! ... Read more

Book Description

Customer Reviews (9)

Counterexamples in Topology
I have found this book to be confusing to use and therefore of little to no value. If I had seen in a bookstore and not Online I would not have purchased it. I also purchased Schaum's Outline of General Topology which is very good.

a veritable mine of information....
To paraphrase Chandrasekhar's review of Watson's Bessel functions text, this is "a veritable mine of information... indispensable to those who have occasion to use point-set topology." I don't think this book is intended to be a text (& I think the authors say so), in which case it would be terrible because it doesn't explain the concepts very much. It's mostly a catalogue of every kind of set you can come up with, every kind of topology you can put on it, and what properties it has such as what T_i axioms the space satisfies, whether it's compact, para compact, etc etc. Most of the time such things are proven, but be prepared to think hard sometimes about the proofs or fill in details. I'm the kind of student where I have trouble understanding things which are highly 'counter-intuitive' so I had trouble proving things, even when I knew definitions, when I did topology for the first time last term. Once I saw this book though I got used to all the weird things in topology (like the ordered square, R in the lower-limit topology, Sorgenfrey plane, etc etc). This book is incredibly useful as a reference.

Essential if you want to be good in point set topology
A distinct characteristic of point set topology is that it builds on counterexamples. If you thumb through any PST text, many theorems are in the form "If the space T is A,B,C, then the space is X,Y,Z". The point of point set topology (pun unintended) is too determine what A,B,C are, and to weaken the hypothesis. "Can we take condition B out? Maybe hypothesis C can be weaken considerably?" How can we answer these questions? You're right, by counterexamples. Students who want to master point set topology should know the various counterexamples, no matter how contrived or unnatural they seem. While textbooks usually present a counterexample to show why Theorem Three Point Five Oh will not work on a weaker assumption -- most students (and teachers) tend to skip these parts. A collection of counterexamples presented in this book (excellent organisation, by the way) is an essential supplement of a topology course; it enables one to 'see' between the points, so to speak.

a good book to combine with a regular textbook
This book has examples in it that are "missing", so to speak, from many regular topology books. It aims to shore up some of these shortcomings, with examples that the student can see and understand. There are charts and graphs, as well as a detailed explanation. Some "problems" often found in regular topology books are solved. Very few proofs, if any, are given. This is not a book meant to be studied without a regular textbook on topology, only to be used as an overall review of problems and short basic premises of topology. Use this in addition to your regular fare, but keep it close at hand when doing homework or preparing for an exam.
There are fundamentals on Cantor's Theorem, the countability or uncountability of sets, compactness, closed and bounded functions, open sets, continuity, connectedness, etc. All these are basic to topology, and this book does address them, but in a brief way. It then shows a basic overview of topology that helps greatly to understand the different fields of topology.

Great Book
As a graduate I encountered a book called "counter examples in analysis" which I found very useful.I alwaysdreamed of such a book in topology, this book exceeds my dreams. It is great.It does not cover all the examples that I have used over the decades but it does cover some that I have never seen.The style is quite readable for a professional topologist.The book goes into a lot of interesting details (and some while not interesting to me would be another person).In short for me it is an essential book.The question is to whom else would this be interesting to. It is clearly of little use to a first year student and less to more advanced student.It's brand of topology is not the current cutting edge.So the audience for this book is limited to a small group and for these people it is top notch. ... Read more

Book Description

Customer Reviews (7)

A masterpiece
First a caveate: This book may not be the most suitable for everyone that takes a FIRST course on General Topology unless he or she is prepared to put in quite a lot of work. This is because the book contains so much information in relatively few pages that the material is necessarily quite dense. Even so the book is a good purchase because it's cheap and will serve everyone good later as a reference.

The organization of the book: Everything is presented in a perfectly logical order, beginning with a summary of Set Theory and ending with topologies on Function Spaces. During the course the reader is invited to make excursions to other areas of mathematics from a topological point of view and perhaps gain insights into those fields that even specialists don't have. This is mostly done through problems for the reader to solve.

Definitions and Theorems: The definitions are always the most general possible, often presented as a set of axioms that the defined quantity has to fulfill. The theorems are almost always presented in their most general form.

The Proofs: The proofs are generally on either the shortest and most elegant form possible, or taken from the original publications. This is for the benefit of the reader even though it might appear to some readers as "terse" proofs because this kind of proofs is the one that gains the reader the most insight once they are understood. "Short and elegant" does NOT mean that the author leaves out details (unless they are explicitely assigned as problems).

Explanations and Motivations: The text is short and to the point. This again does not mean that the author leaves out anything relevant or that he does not warn for possible pitfalls.

Examples of introduced concepts and definitions: There are numerous well chosen examples, often nontrivial, to illustrate the meaning of introduced concepts.

The problem set: The set of problems is just fantastic. The problems are numerous, diverse, illustrative, and again, sometimes HIGHLY nontrivial. Don't be too scared though, because the author provides very accurate hints of how to approach the more difficult ones.

Bibliography, Historical Remarks and Index: One just has to admire the amount of work the author has put into this.

Miscellaneous: As mentioned, the material is (necessarily) condensed, but the text is never "dry" or boring. There is an undertone of humour in quite a few places. For instance, when the author mentions that not every regular space is completely regular, because there exists a formidable example that shows this fact, he relegates that example to problem 18G "where most people won't be bothered with it". This practically guarantees that most people WILL be bothered by it by looking up 18.G. There, in 18G, he provides som many hints that it is actually doable for most people to reconstruct this formidable (i.e. difficult) example.

On the Downside: There are no solved problems, and the author does not teach the reader on HOW to solve problems. This is however compensated for by the numerous hints in the problem set and through the methods of thaught one learns from reading and understanding the proofs. Also, in topology, one basically has to invent ones own mothod to solve an unsolved problem. There is no canonical way of doing things!

Excellent
this is an amazing book. very wisely constructed with a lot of real content.
if i may ask for something more i would ask for an updated version, and solutions for problems.

Topology encyclopedia
Every mathematician need this book if he want to learn topology. Is a classic. You can use it for learn and as a reference. Further, this book has a extensive bibliography.

Wilard's Topology
After looking at several books on topology, I would have to say that Wilard's General Topology is an excellent resource book.For those who have taken a topology course and want a little more practice with problems, this book has numerious exercises that help form an solid knowledge base.What else is nice about the problems is they are good research-starters for undergraduates.The examples in the chapters are non-trivial and explain the ideas of the chapter.Also, Wilard's General Topology has a slight set-theoritic view to topology, so those who like set-theory and topology, this book will be of great use.I suggest Wilard's General Topology if you need another topology book to help explain ideas from class or other books.

Very terse
I found this book very terse.It seems like it could be a great book for someone already familar with topology, I don't recremend it for anyone who is not very familar with the subject.It lacks examples of how to solve problems, then throws quite complicated ones at you very quickly.A few mistakes in printing also. ... Read more

Book Description

Customer Reviews (6)

Intro to Topology
This is a wonderful book to start your topological studies with. It has manyproblems for one to do so one can practice and study and have the ability to make the grade on one's tests

Great book on Topology
I bought this book for my own enlightenment after already having a course in Topology here at Penn State University.What I find most interesting about this book is that the author explains the philosophy on the ideas and what we are really trying to say with these definitions and theorems.The book I used in my course didn't explain much at all so it would have been much more difficult to teach yourself from this book.Topology is somewhat abstract so if you're looking to study Topology this is a great book to start.A word of advice, read over a theorem and proof and try to reproduce it on paper from your mind.Help yourself from the book a bit along the way if necessary.You will learn much more this way as opposed to following along the proofs in the book as you read.You might also be interested in Counterexamples in Topology, a book with thousands of counterexamples.

Excellent Text for the Price
I chose this text for an independent study course in topology, but I ended up switching to Munkres fairly quickly. Considering the price it is an excellent text. All the fundamental topics of point-set topology are covered in a clear and orderly manner, but Mendelson treats metric spaces in much detail before he ever gets to definition of the topology. Some people may prefer this approach, but I think the definition should be introduced first.

Good Introduction to Metric Spaces and Topology
I was not a mathematics major, and only in recent years have I ventured into abstract mathematics. I was motivated to learn about topology as an aid to understanding a particular 3-D earth modeling application.

I read Introduction to Topology in three stages: as a review of set theory and metric spaces (chapters 1 and 2), then as an introduction to topology (chapter 3), and lastly as a detailed look at two important topological properties, connectedness (chapter 4) and compactness (chapter 5). I had previously read (and reviewed) another book titled Metric Spaces by Victor Bryant, but Mendelson is my first serious look at topology.

My reading of Mendelson - a 200-page text - required about 100 hours, substantially longer than the 40 to 60 hours estimated by an earlier reviewer. No solutions are provided for the section problems, which are generally proofs, not explicit problems.

The first chapter provides a concise overview of set theory and functions that is essential for Mendelson's later chapters on subsequent set-theoretic analysis of metric spaces and topology.

The second chapter is a solid introduction to metric spaces with good discussions on continuity, open balls and neighborhoods, limits from a metric space perspective, open sets and closed sets, subspaces, and equivalence of metric spaces. Chapter 2 concludes with a brief introduction to Hilbert space.

The third chapter introduces topological spaces as a generalization of metric spaces, and many theorems are largely restatements of the metric space theorems derived in chapter 2. I was thankful for this approach.

Mendelson begins chapter 3 by demonstrating that 1) open sets and neighborhoods are preserved in passing from a metric space to its associated topological space and 2) the existence of a one to one correspondence between the collection of all topological spaces and the collection of all neighborhood spaces.

He then reminds us that in a metric space we can say that there are points of a subset A arbitrarily close to a point x if the metric d(x, A) = 0. In characterizing this notion of arbitrary closeness in a topological space, Mendelson introduces the closure of A, the interior of A, and the boundary of A. Other topics included topological functions, continuity, homeomorphism (the equivalence relation), subspaces, and relative topology.The final sections in chapter 3 on products of topological spaces, identification topologies, and categories and functors were more difficult.

In chapter 4 the initial sections (connectedness on the real line, the intermediate value theorem, and fixed point theorems) were largely familiar. But thereafter I became bogged down with the discussions of path-connected topological spaces, especially with the longer proofs involving the concepts of homotopic paths, the fundamental group, and simple connectedness.

Chapter 5, titled Compactness, was even more abstract and difficult, with topics like coverings, finite coverings, subcoverings, compactness, compactness on the real line, products of compact spaces, compact metric spaces, the Lebesgue number, the Bolzano-Weierstrass property, and countability. Perhaps, a reader more familiar with analysis would have less difficulty with the last two chapters.

In summary, Introduction to Topology is quite useful for self-study. Mendelson's short text was intended for a one-semester undergraduate course, and it is thereby ideal for readers that either require a basic introduction to topology, or need a quick review of material previously studied. The last two chapters on connectedness and compactness are substantially more difficult, but are still accessible to the persistent reader.

Ideal for self-study
This book is ideal for self-study. If you have not had the luxury of taking a topology course during your undergraduate studies, but you need to know some topology and you have to study it by yourself, this is the book you need. It is very readable and it explains carefully every concept. However, it is just an introductory text and it contains only basic material. You don't have to invest a lot of time to study the material in this book: let's say 40-60 hours of study are enough to grasp everything. I reccomend it especially to those graduate students of applied mathematics, finance, statistics or economics, who need to use some basic result from topology in their work. ... Read more

Book Description

Taking a direct route, Essential Topology brings the most important aspects of modern topology within reach of a second-year undergraduate student. Based on courses given at the University of Wales Swansea, it begins with a discussion of continuity and, by way of many examples, leads to the celebrated "Hairy Ball theorem" and on to homotopy and homology: the cornerstones of contemporary algebraic topology.

Customer Reviews (2)

Best undergraduate topology book
I have never seen such a beatiful explanation on continuity and its relations to series and sets. Now I understand why, when mathematics is lousily explained,everything seemms to be so hard. I recommend strongly this book for someone for self study on topology. Hope the author can write on other topics of mathematics.

A pleasure to read
I have a major in math, many years ago. I have moved into economics, but miss the elegance of math, hence I decided to revisit some old topics, and started with topology. As a student we used lecture notes and no real textbook, so my choice now was this textbook. It is a pure pleasure to read. I wish we had used it as a text book when I studied.

The topics are well motivated. Crossley does a good job in explaining why we should care about these particular lemmas and theorems. The proofs are usually elegant. I find the estetic pleasures a good math book should provide. ... Read more

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

Customer Reviews (13)

One of the worst textbooks ever.
The only good thing about this textbook is that it contains possibly every topic that an instructor could conceive of covering in an introductory course in algebraic topology.However, there are several flaws which prevented me from getting much out of this book:

1. Lack of definitions:I agree with another reviewer of this text who stated that this book does not give definitions.It often does not define objects which not only have several equivalent definitions, but it is not even clear which definition is being used.I often had to search through several books to find a definition which seemed appropriate in the context of this text.

2. Useless or non-existant examples:When a definition is given, it is rarely followed by an example which illustrates it.When examples are present, they don't clearly illustrate the definition.It would be much better to have one well-written example for each definition (or at least for many definitions) rather than five examples for the same definition, none of which illustrates what each part of the definition is.

3. "It is clear":This phrase is used to such unreasonable extremes in this book that it makes me wonder why the book was written at all.If everything is so obvious, then why did you need to write a book about it?Algebraic topology is an inherently abstract and difficult subject, and the author should not have assumed that everything would be so obvious to all parties reading his text.

This book does have another redeeming quality: lots of exercises.Once you finally do understand the subject matter (whether due to a good instructor or a better reference textbook), there are lots of exercises to put your understanding to use.

Excellent book for geometers
I have taught graduate algebraic topology courses three times from this book.My overall feeling is that, despite a few flaws, I have not seen another book I would rather use -- and I really wish this book had been around when I was learning the subject!I appreciate its very geometric style and the way it tries to get the reader to "see" the definitions of homology, homotopy, etc, before diving into the rigorous treatment.Many algebraic topology books I have seen are nearly example-free; they build the theory but don't show the reader how to do much with it. In contrast, Hatcher spends a lot of time, appropriately, on some of first really important examples in topology, such as surfaces and projective spaces.These investigations are continued in the exercises, which I feel are the best thing about this book.Many of them contain juicy examples which really show how the geometry and algebra interact.

On the minus side, I would agree with another reviewer that sometimes the rambling style, which works quite well in the introduction to a new concept, sometimes gets in the way when it's time to get down to precise definitions and theorems.

Definitely a Bible
This is certainly a modern classic that predominates algebraic topology courses like 18.905/6 at MIT and Part II and III Mathematical Tripos at Cambridge. It is also perfectly suited to personal study and reading -- savor it and its geometric beauty! I would warn the absolute beginner that the text may seem steep at first (especially if you start with Chapter 0 first, the beginning of Chapter 1 is easier) and slightly unmotivated. I would recommend Massey's "Algebraic Topology: An introduction" GTM 56 for preliminary or complementary reading. Be warned that the styles are very different. Hatcher as well as Munkres like introducing the Fundamental Group pretty much right off the bat, which I like. However, there is something to be said to getting to beef up your geometric intuition by thinking about projective space and learning some classification theorems about compact manifolds and this is the approach of Massey. Massey is also nicer if you have just finished a first undergraduate course in topology.

Hatcher is definitely every algebraic topologist's bible and this really isjust volume one in a whole series (check Hatcher's Cornell website for more info) of books that will be as monumental as Spivak's 5-volume Comprehensive Introduction to Differential Geometry (Which you should also buy as each volume is only ~$40). We should take a moment to pause and appreciate what Allen Hatcher has done by putting the book online for free. This is a tremendous statement that learning and knowledge should be free and accessible to anyone who seeks it. I know I first printed Chapter 0 out and starting reading it for free, but to be honest the quality of printing and binding done by the Cambridge University Press is worth the 30 bucks and you should pay it to keep academics warm and off the streets.

Bible of Algebraic Topology
You can not find a better book that explains and covers this beautiful subject better than Allen Hatcher's Algebraic Topology.The subject is build up very well and there are tons of examples that will help you deepen your understanding.I read this book in parallel with Sato (Algebraic Topology: An Intuitive Approach) and Munkres (Topology, 2nd Edition) for independent study.This combination is working well for me, but don't expect to get the same results as you would if you had a great teacher.

Mixed Feelings
This book is intended as an "introduction to alegbraic topology" and I rated the book accordingly.

I found the book refreshing at points and thorougly frustrating at other points. This was one of the first book I approached when trying to learn formal algebraic topology. Prior to reading it I had indirect exposure to algebraic topology in application to physics especially when learning about differential forms where one is usually exposed to homology cohomology and derham cohomology, etc. I found the physics texts MUCH more instructive than this text which is supposed to be from the mathematicians perspective.

The book has it's merits:

1) it is organized well and attempts to relate the main topics in algebraic topolgy - homotopy and homology
2) it has many examples to help solidify the concept presented
3) it has plenty of exercises of varying difficulty.
4) it genuinely tries to motivate the mathematical ideas of algebraic topology.

However it has many faults. I was particulary disturbed by it's lack of definitions. At some point I felt like I was having a conversation or reading a "pop" math books for the dilettante not mathematician. I found myself repeatedly going back and having to REREAD THE TEXT to get the definition of some mathematical object. In my humble opinion a math text should clearly state definitions and properties and not try to "explain" them in prose without the preceding definitions.

The author also states minimal prerequisites ( algebra and point set topology ), however, it is clear alot more is needed.

Although there are plenty of examples, the author, simply states conclusions which maybe "self-evident" to someone with previous exposure to algebraic topology but not to a novice. In the examples little effort is made to explain the assertions.

Finally, the author has a chapter 0 which goes over some geometric preliminaries with little rigour, which to his credit he admits. However, he states that you do not really need to read it thru and only refer to it as needed when going over the text. The problems is all of the notions used in chapter 0 are ASSUMED TO BE KNOWN in the text. You have to know all the constructions, definitions and properties and access them from memory at a moment's notice to follow along the proofs and examples. That is not difficult to do but he doesnt present these notion in chapter 0 in a clear and efficient way. Again it is presented in "prose" format.

Regardless, I suggest you download the electronic version and read it for yourself. Google the author and the link will pop up.

I wanted to rate this book a B- but there was no 3.5 so I gave it a 3. ... Read more

Book Description

Book Description
In this broad introduction to topology, the author searches for topological invariants of spaces, together with techniques for calculating them.Students with knowledge of real analysis, elementary group theory, and linear algebra will quickly become familiar with a wide variety of techniques and applications involving point-set, geometric, and algebraic topology.Over 139 illustrations and more than 350 problems of various difficulties will help students gain a rounded understanding of the subject. ... Read more

Customer Reviews (10)

Bad, Bad Book
This book is terrible. The author doesn't denote important material at all! Sometimes the most important part of a section is contained in one poorly written sentence. This book is subpar. Both Munkres and Hatcher provide everything this book does, in fact much more so, and presents the material in much more rigor. I haven't seen a worse introductory book on the subject, though for some reason people who already know the material seem fond of the book.

I'd give it -5 stars if I could.

A very welcome, intuitive approach to topology
Many of the standard introductions to Topology (Munkres comes to mind) focus more on the logical flow of the material, and less on the motivation for the material.This book focuses on the motivation, but after the first few chapters, the logical development is sound too.

The Armstrong book starts out with some fairly advanced concepts, outlining some interesting topological results before giving the modern definition of topological spaces in terms of open sets.Typically, authors give the open set definition of a Topology at the outset, before explaining what topology really is, and without explaining why that definition is used or how it was developed.Armstrong instead shows the historical motivation of the subject, and actually leads the reader through this development, starting with the less elegant but more intuitive definition of spaces in terms of neighborhoods.The equivalent open set definition is then taken in chapter two.However, once things get going, this book does not move slowly at all--quotient spaces and the fundamental group are presented early and covered in depth, and it is not long before the reader encounters genuinely advanced material, in rigorous form.

It's true that this book doesn't cover the same amount of raw material that a book like the Munkres does, and it's true that the book does not follow the most concise logical order, but it offers history, motivation, and initial exposure to more interesting results.Perhaps more importantly, it develops the reader's intuition.In many ways, this book is a complement to the Munkres, and an enthusiastic self-learner would benefit greatly from using both books simultaneously.

At the same time, this book does get into some more advanced topics.It has a particularly clear exposition of simplicial homology.My last word of praise about this book is that although it gives lots of motivation, it is still very concise.I think it's hard to go wrong with this book.

Your Average topology student will be frustrated...
This text is very very difficult to read for people like me, your average topology student. A difficult subject to grasp, the layout of this book simply does not help organize the material. I have purchased several other books, that while they don't make topology easy, at least make it digestable. Pass on this book and go with Munkres.

An acceptable text
I would recommend reading with a highlighter and marking up a lot of the text because many definitions, points of interest, etc... are not set apart from regular text and it can be difficult locating the information you want/need to know on a particular page because of this. I have already highlighted a good deal of the book so that I can flip through the pages quickly and locate what I need.

There are plenty of exercises in the book of easy to medium difficulty, but certainly not many that I would call "hard."

The text is easy to read even if it is not organized as well as Munkres book. I don't think this is a book anyone would regret getting for learning topology for the first time, but as the title clearly indicates, this is not a book for people taking a second course in topology.

NO NO
The fact that the author does not explicitly define things is a bad enough reason to stay away from this book. If you only want a light treatment of point-set topology, go for Munkres, otherwise, Hatcher. ... Read more

Book Description

Customer Reviews (6)

I'm not good at math
I wanted to teach myself some topology and a friend with mutiple Math PhD's reccomended this book to me.
This is a tremendous value, and is comprehensible. But it is prety lean and direct, so be prepared to work on this in a quiet place where you can concentrate for a sustained period of time. Proofs are direct, and expect you to be familiar with notation through all of Algebra.
I re-emphasize: there is zero, no, nada, blank, null coddeling here. Every single word, every single notation is important, and if you haven't read, marked, and inwardly digested each one it is a promise you will be lost in a page or two and have to go back. There is no fat here at all and the authors don't babysit you or expalin anything five different ways. This is direct on the coal face math.
Still, I knew only basic basic basic totpology before this, and now I have a vague understanding of all the major areas of further inquiry.
A very good value.

Okay, not great. Overall, I give it a C+
The exposition, while clear and not without attention to subtleties of the theory, is a little scattered. The metric spaces chapter is very good, but after that, it goes downhill. In particular, I was pretty disappointed that the mean value theorem was not proved as an application of connectedness. Everybody sees the mean value theorem in calculus, and the proof is really quite elegant. Also, a lot of important notions in topology are relegated to the exercises, and the rest of the exercises are like applications to analysis. It's kinda nice to be challenged to see the definitions in multiple ways through the exercises, but it would be nicer to get all the perspectives in the exposition, and be given exercises that would deepen one's understanding of the material.

All in all, this book feels like "topology as a branch of analysis" and only helps the reader to develop a modest working topological intuition. For readers interested in topology as its own subject, Munkres' book is the only book. For those readers desiring a more introductory approach, I found Mendelson's book to be an excellent introduction - the chapters on connectedness and compactness are thorough and quite helpful - though that book is lacking in that it doesn't discuss separation axioms at all, and contains few exercises. But that book is unique in that it despite its brevity it touches on metric spaces, categories, and the fundamental group.

If you're going to read this book, get a copy of Mendelson's book - it will flesh out your understanding of topology.

excellent introduction to topology
I used this book to teach myself the basics of point-set topology and homotopy theory. What makes this book so great is that the author doesn't waste words in delving into the heart of a concept, while providing insight into it. A good collection of interesting problems, most with solutions in the back of the book. This makes this book very good for self study. If you liked Rudin, you'll probably like this book as well, as it is written in a similar style. If someone knows of a better introduction, do let me know.

exceptionally well organized
This is a lean fast introduction to topology at the third or fourth year level. Pure math types only. The book is terse but the topics are selected with care and one things leads to the next. The proofs are sufficiently detailed. Nearly every exercise has a solution in the back. The clearest exposition of the fundamental group I've seen.

Good grad school prep.
This is the usual text for introductory Topology at UCLA, where I took the course.The authors (who teach at UCLA) have "if you haven't chewed through every syllable you are not learning" mentality.In short, the book is terse and demands a lot from the reader.Looking back, this was great preparation for graduate school and is probably the best philosophy for the serious undergrad.The book contains all of the information one needs for an introductory course, but absolutely no more.Not a single character is wasted on "extraneous" explanation.Be ready for battle when opening this one, but it's worth it. ... Read more

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

Customer Reviews (4)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Customer Reviews (1)

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

Book 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 the material accessible to undergraduate students without requiring extensive prerequisites in upper-level mathematics. 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 exposes students to the exciting world of topology now(!) rather than later. Students using this textbook should have some exposure to the geometry of objects in higher-dimensional Euclidean spaces together with an appreciation of precise mathematical definitions and proofs. ... Read more

Customer Reviews (2)

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

Book Description

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 intended as a textbook for a first-year graduate course on algebraic topology, with as strong flavoring in smooth manifold theory. Starting with general topology, it discusses differentiable manifolds, cohomology, products and duality, the fundamental group, homology theory, and homotopy theory. It covers most of the topics all topologists will want students to see, including surfaces, Lie groups and fibre bundle theory. With a thoroughly modern point of view, it is the first truly new textbook in topology since Spanier, almost 25 years ago. Although the book is comprehensive, there is no attempt made to present the material in excessive generality, except where generality improves the efficiency and clarity of the presentation. ... Read more

Customer Reviews (6)

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

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

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

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

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

My secret weapon in topology
Today I told someone that Bredon's "Topology and Geometry" book was my secret weapon.I say this because it has mostly everything a grad student in topology needs to know in order to be fluent in the subject.Ibought it not because I was taking a class from it--I got it because I sawit at the library and realized that this is the 'end all' topology book forme and that I will not need to buy all those 'other' books in topologyanymore.I will warn you though--some of the proofs are terse, so you haveto be somewhat 'mathematically mature'.It is definately harder thanMunkres' "Topology: A first course" (the red book).But it goesbeyond the point set crap that gets (in my opinion) too much airtime.Iwish I would have taken a class under this book. ... Read more