Reviews (6)

Terrific Book
Enderton's writing is the best I've seen in any introductory math textbook; he is lucid, well organised, comfortably paced but free of expository flab. The exercises (judging from chapters 2 and 3) are not terribly difficult, but quite useful in building one's intuition and connecting logic to other mathematics. I had the book for my Logic class as a first-semester sophomore with very little experience with proofs and no abstract algebra, and found it quite accessible. I guess the book starts off with an advantage, being about a subject as interesting as logic, but that does not seriously detract from its merit.

Very Good.
I read the FIRST EDITION. This is definitely the best introductory mathematical logic book I've seen. It's the most rigorous, most advanced (a reasonably strong form of Godel's theorem is given), and is well-organized and very clearly written. It would be suitable as 1)an introduction for students with some mathematical experience- say a little abstract algebra and perhaps some previous exposure to logic. 2)a refresher for advanced students 3)a nice reference for basic topics. The exposition is great- Enderton always clearly explains what he's doing and why, keeping the reader focused on the big picture while going through the details. He helps to place topics in perspective, and has organized the book so readers can skip some of the more involved proofs and sections on the first reading. Chapter 1 covers propositional logic, with a general-purpose discussion of inductively defined sets, unique readability, and recursion. Many books these days do a sloppy job justifying recursive definition, or dont bother at all- Enderton does it right and is fairly detailed. Chapter 2 begins first order logic and has the most detailed proof of the completeness theorem I've seen. Sect 2.7 concerns translating between theories in different languages, something i hadnt seen developed explicitly before. 2.8 is a great exposure to nonstandard analysis- long enough to give you an idea how it works and why its useful. Chapter 3 begins with an analysis of some reducts of number theory- (N,0,S) (N,0,S,<) and (N,0,S,<,+) and shows how to eliminate quantifiers in them. Next, toward Godel's theorem, a finite set of axioms for a subtheory of number theory is given, and a host of relations and functions are shown to be representable in this theory. In 3.5 we get the fixed-point theorem, Tarskis thm, a weak Godels thm, a stronger Godels thm, and Church's Undecidability thm, and an introduction to the arithmetic hierarchy. 3.6 lifts Godels thm to show set theory is incomplete, and discusses Godels 2cd thm. Chap 4 is 2cd order logic, skolem normal form, many-sorted logic (a first order logic with different sets of variables ranging over different universes), and general 2cd order logic (restrictions are placed on the subsets "X" ranges over in the 2cd order formula \all X \phi). Basic recursion theory is developed throughout the book- Enderton begins with informal notions of computation, then defines a relation R as recursive iff it is representable in some consistent finitely axiomatizable theory, and discusses Church's thesis. 3.8 quickly covers universal computers, partial functions, Kleene normal form, unsolvability of the halting problem, the smn thm, Rice's them, and a register machine model. All this seemed a bit disorganized, so familiarity with computation and automata theory would be a plus. Heres the contents for the first edition, c1972:

Chapter Zero - USEFUL FACTS ABOUT SETS. . . .1
Chapter One - SENTENTIAL LOGIC
1.0 Informal Remarks on Formal Languages 14
1.1 The Language of Sentential Logic . . . . . 17
1.2 Induction and Recursion . . . . . . . . .22
1.3 Truth Assignments . . . . . . . . . . . .30
1.4 Unique Readability. . . . . . . . . . .39
1.5 Sentential Connectives. . . . . . . . . .44
1.6 Switching Circuits. . . . . . . . . . . .53
1.7 Compactness and Effectiveness. . . . . 58
Chapter Two - FIRST-ORDER LOGIC
2.0Preliminary Remarks. . . . . . . . . .65

2.1First-Order Languages. . . . . . . . . .67
2.2Truth and Models. . . . . . . . . . . 79
2.3Unique Readability. . . . . . . . . . . 97
2.4A Deductive Calculus . . . . . . . . . .101
2.5Soundness and Completeness Theorems. .124
2.6Models of Theories. . . . . . . . . . . 140
2.7Interpretations between Theories. . . ... 154
2.8Nonstandard Analysis . . . . . . . . . . .164
Chapter Three - UNDECIDABILITY
3.0Number Theory . . . . . . . . . . . . 174
3.1Natural Numbers with Successor. . . . 178
3.2Other Reducts of Number Theory. . . . 184
3.3A Subtheory of Number Theory. . . . . . 193
3.4Arithmetization of Syntax. . . . . . . . .217
3.5Incompleteness and Undecidability . . . 227
3.6Applications to Set Theory . . . . . . . .239
3.7Representing Exponentiation. . . . . . .245
3.8Recursive Functions. . . . . . . . . . .251
Chapter Four - SECOND-ORDER LOGIC
4.1 Second-Order Languages . . . . . . . . . 268
4.2 Skolem Functions . . . . . . . . . . . . 274
4.3 Many-Sorted Logic. . . . . . . . . . . . 277
4.4 General Structures . . . . . . . . . . . . 281
Index . . . . . . . . . . . .291

Reviews (3)

Sloppy, with gross errata.
The two stars are for the one good feature this book has: an extensive table of symbols, listing the first page of text on which the symbol appears in the book.This rare feature could well be adopted by all math textbooks.But sadly, while a meaning is linked to each symbol, the grammar of the concatenated symbols is not addressed, to the continued detriment of the reader.There are some really sloppy errors, even in the exposition of axioms, where precision matters most.Very poor print quality. Outdated and massively ovepriced.

An Excellent Introduction
Perhaps because it is a Foundations book -- in my mathematics training it always seemed that the people who did the best job of motivating and explaining (or at least making you feel you understood) the material were Foundations people -- but this book has a presentation polished to the point where the closest genre of mathematics text in level of polish would be intro calculus books, where the problems theorems and proofs have been worked over for many many many years.Here, however, the material is in great part relatively recent - probably the closest to contemporary stuff you can see as an undergraduate -- in Real Analysis, by contrast, you may well just be coming out of the 19th century by graduate school.This polish, I have discovered in later years, facilitates use of this book for self-study and it is a wonderful text for providing rapid refreshment of important concepts.I have over the years referred back to it on a number of occassions and have always been pleasantly reminded what a wonderful book it is.
This is a very nice book and the best introduction to the material I have seen (although, given the number of intro books I have seen on the topic, this may not be a strong statement).

Reviews (27)

The book after Herstein
I think I would only recommend this book to someone who has already had some exposure to algebra (or one especially gifted in mathematics). The beginning of the book is not too bad, but towards the end of Part I the pace quickens quite a bit. If you are willing to read over the text many times, and do all of the non-trivial exercises (there is an impressive olla podrida of algebra in them, most of which are the beginnings of some very deep ideas), then it should be a very rewarding experience. Namely because this is one of the most readable textbooks which covers everything from groups, rings, and fields to homological algebra and algebraic geometry. It is very rare to see this much material covered in one book, and for it to remain so structured (Rotman is an example of a book that covers a lot of material, but loses its structure somewhere).

Excellent book
I am surprised that this book has not got the 5 stars. It is very suitable for advanced undergraduates/first-year graduates. The book is full of examples; and the proofs are amazingly clear and succinct. The book introduces new concepts in the excercises long before the student encounters them in the sections.

This is a beautiful way to teach mathemtatics,--and indeed to learn it. The book is replete with examples that connect concepts from toplogy and real analysis with Algebra.

This book definitely deserves the 5 STARS.

Comprehensive book
I'm a graduate student in math. We used this book for the basic year-long abstract algebra sequence: group theory, chapters 1-4 and some of chapter 5; ring/field/galois theory chapters 7-9, 13-14. Some of my fellow students took a module theory course which was at least partially based off chapters 10 and (I think) 11. I'm sure more advanced courses could easily be based off chapters 15-end. Considering the cost of university books, I consider it very nice to buy one book for essentially 3+ courses.

The exercises in some sections are very diverse. My group theory professor made us do a huge number of them, and now I am amazed at how often I see questions similar to those from Dummit-Foote show up on past qualifier exams from many different universities. Regarding lack of answers in the back...well, you shouldn't need too many, and if you get really stuck, that's what the professor is for. And if you're learning it on your own then I'm thinking you should be brainy enough not to need answers!

The text itself is very readable and complete.

I don't think I'd recommend this as an undergrad textbook, although I've no doubt that there are some clever undergrads who could learn from it. I used Herstein's "Topics in Algebra" for my intro-to-abstract course as an undergrad. Herstein is designed to be introductory in nature, though still a wonderful book, while DF is more encyclopedic.

Reviews (6)

A Valuable Book!
I admit that this book might not be suitable especially for pure mathematicians. But I very much liked Silverman's way of writing: He cast questions and encourage readers to tackle them! Indeed, this is a unique number thoery book written in that way.

A reallyfriendly, enjoyable introduction to number theory
I very much enjoyed this book.The book is indeed an excellent and "friendly" introduction to number theory.Dr. Silverman writes in a conversation style.I felt like I had a friendly tutor standing over my shoulder explaining not only how the mathematics worked, but, more importantly, why the topics he described or was about to describe are important and their relevancy in either the world of mathematics or in the "real" world.While he has very few "formal" proofs compared to most number theory texts, Dr. Silverman thoroughly works through numerous numerical examples to give the reader a "feel" for what is going on.

I was particularly pleased with Dr. Silverman's chapter explanation of why quadratic residues are important and how they are used.

Dr. Silverman presents introductory explanations of a number of frequently mentioned number theory topics such as Mersenne Primes, number sieves, RSA cryptography, elliptic curves.He ties together lucid explanations of Pythagorean triples, x2 + y2 = z2, x4 + y4 = z4, and elliptic curvesto build to an explanation of Wiles proof of Fermat's Last Theorem.

This book was NOT written for math majors
I just wanted to make clear the point that each textbook or math book written is written for an INTENDED audience, and it's not fair to negatively criticize a book by using the reviewer's own personal background, rather than the INTENDED audience, as the guide for criticism.

This book was not written for math majors.So, I find it kind of distressing to hear that many math majors are saying this was textbook for a beginning number theory class for math majors.Silverman makes effort to point out that the book was written as the textbook for a general liberal arts math class, which is actually taken by non-science and non-math majors at the university where Silverman teaches.It requires nothing beyond basic calculus (if that), and I don't see anywhere where Silverman gives the impression that the book is meant to be used as a strong introduction to writing proofs or becoming fluent in rigorous mathematical arguments which math majors will later see.

So, of course, math majors will find fault...but the book wasn't written for them.It was written primarily to get people who have little interest in math or little exposure to math, some opportunity to see something more interesting beyond high school algebra and calculus.The emphasis on computation is warranted in any case, because although number theory is mathematics and has rigorous proofs, intuition and working familiarity with the concepts and constructions of number theory only come through hours and hours of simple computations with the positive integers.Computation is a legitimate and necessary part of number theory.

As for rational points on the circle (and Fermat's Last Theorem) being unusual or out of the ordinary material, this is farthest from the truth.The example of rational points on the circle is one of the oldest (2,000 years or so???) and most basic constructions of number theory, revealing how geometric number theory is, and the example directly leads to more general ideas and concepts which are central to current research (Diophantine equations, elliptic curves, projective geometry, for example) and pick up many of the standard graduate references on elliptic curves and the first 5-10 pages are a detailed examination of this very example.

I'm a graduate student studying number theory, so I'm pretty far away from the intended audience.But I can see that the book does a pretty good job at what it sets out to do, namely present an exposition of certain problems in mathematics, accessible to non-science and liberal arts majors, in a leisurely and engaging fashion, and to get the students to do their own basic pattern-searching, computation, data collection and conjecturing (ALL important facets of mathematics...proof is the polished product, but lots of time is spent by mathematicians before even GETTING to the point of proving things.)

Reviews (21)

Wedderburn, Waring and Hamilton
Not necessarily in that order.President McCosh of Princeton
waxes eloquent in his Scottish Philosophy book somewhere on
the internet, re: Dugald Stewart, Kant and Hamilton.

Hamilton is a strong vice, but clearly represented in Herstein.

Good Introduction, useful for self study
I am an engineer by training and a sales man by profession, with a a strong liking for mathematics.
I found this book to be an very readable introduction to a subject (abstract algebra), I had never been exposed to during my engineering math - other than matirx theory, which was obviously taught extensively.
The proofs are generally easy to understand, but certainly not trivial.
A pleasure to read

Reviews (15)

Wonderful Abstract Text
This is possibly the most addictive Abstract Algebra book ever. Filled with a wonderful selection of problems, Gallian discusses many aspects of groups and rings in a way that is enjoyable to study. The love of mathematics that is shown in this book is reflected by how Gallian eloquently weaves principles of group and ring theory with beautifully selected INTRUIGING problems. I enjoyed this book more than any other math text I have had.

Great Book!
I'm in a 4th year group theory class and have found this book to be highly useful in learning my course material. It gives lots of proofs, lots of excercises with lots of theory and good explanations. I have no complaints!

Gallian's poor algebra book.
I agree with the other reviewers in the sense that it is ture Gallian's book is soft on theory and rigor, but oppositely I find this lack of real substance to be Gallian's deepest flaw. I give Gallian one star, basically for effort.

I divide my critique into the following subcategories:

Organization:

Gallian's book is organized well enough in the sense that he opens each chapter with some commentary about the problems to be studied, or motivation, and then proceeds to go example, theorem, proof, example, example, example, example,..., example. This doesn't work, I think, because he spends too little time actually showing theorems and proofs, and sometimes he'll build an entire chapter on just two or three theorems, and fill the rest with useless commentary (which I'll mention again below).

Readability:

As for readability, for people who read math books at all (i.e., those who study outside of class), this book should be a nightmare. If you were to strip away all of the useless commentary/endless biographical insets/weblinks you would be left with probably about 30 pages of theorems and cumbersome proofs (by cumbersome, I don't mean involved, I mean unrefined). Gallian has failed to make a readable text because he presumes to have the omnipotence and foresight required for putting a full understanding of algebra and algebra history into one book. As a result, the excess commentary he makes and useless statements (for example, "In high school, students study polynomials with integrer, rational, real, and sometimes complex coefficients") distract a reader from the main points, and I rarely found myself rubbing my chin thinking how insightful something he said was. All in all, I feel as though the reading felt "hoakie" at best--like he was elbowing me in the side, winking, trying to get me to lie and say I thought what he was saying was insightful.

Exercises:

The exercises are often clumsily put together and the quotes before each problem set can get extremely patronizing. I remember thinking how cocky this Gallian fellow must be to presume that people can't do "his" problems. A joke, to say the least. In any case, they seem fine for all purposes -- if you're going into chemistry or an applied science that uses group theory. It's very obvious that our author believes that group theory is the pinnacle of the algebra experience and struggles to present topics from rings and fields. IF you are someone who likes group theory, fine. BUT Artin's book does everything Gallian does and more with group theory and builds the same ideas on more solid footing, using linear algebra excessively throuhout the book. For example, if you think I'm joking about Gallian's weakness, just look at the chapters on isometries and compare them to the chapters in Artin, and you'll see what I'm talking about.

Peter Rabbit:

Well, I do have at least one nice thing to say. As anyone can see, Gallian has a lot of examples, but this seems to be the only redeeming quality of the book. But that alone doesn't make an algebra book.

Broad Commentary:

Reviews (19)

Medium-level Book: Use only as a supplemental
The last edition of this book was my first exposure to this amazing topic.My experiences with that edition, however, were not very favorable.The pedagogical style was Byzantine and not well laid out.Many key ideas of the text were introduced in the exercises and never fully developed in the main text itself.Despite being highly verbose at times, ironically little detail and clear exposition was given to illuminate either the motivation behind a topic or proof or such presentation in a succinct, clear fashion.Moreover, many of the key, standard notations used in modern abstract algebra were also blaring absent or hidden, again in the exercises instead of explicitly emphasizing them in the main body of the text.This in itself made for a less than ideal first exposure, which was made all the more complicated by our instructor Linda Chen (Ohio State University).She seemed barely familiar with the poor nature of the book, suggesting that we use Gallian or Artin as "supplemental" readings.I wish she had chosen Artin, a book of great clarity and maturity in both the treatment of the subject and the reader as well.Gallian, in contrast, was even more obtuse in being ever more verbose in the proofs compared to Fraleigh.Adding more to the confusion was the fact that Ms. Chen largely made up her own problems, which often proved flawed with errors that made some of the assignments outright intractable.She would assure us "this is a difficult subject and you'll need to spend more time with it."Apparently it is a difficult subject, so much so that she herself had problems with both mastering some of its subjects and teaching it to others.This was underscored dramatically when one of the students came a couple of minutes late to class and in the middle of her proceeding to explain how to solve one of her bogus problems.He sat there quietly and listen and then cleared his throat, gently interrupting her, stating that the problem was in fact unsolvable and the only thing you cold prove about it was that it was unsolvable.She then handed him the chalk incredulously to do just that.And he did, much to her red-faced embarrassment and our great amusement.I mention this not condemn Ms. Chen, but to offer sympathy to those that have had such similar experiences.All it not lost.Many, including some truly brilliant professors, have had similar problems in their undergraduate programs and didn't do so well the first run with this subject either.I truly didn't like this topic at first, but have now come to love it and am never disappointed in the freshness of its richness, finding it in a broad range of topics.This new edition of Fraleigh, seems a much greater improvement, but I would suggest it be used as a "supplemental" text.Instead Herstein, Artin, or Robinson should be used.Herstein and Robinson are much slimmer books, succinct, and the the exposition direct and clear.I learned in a couple of months more than I did in nearly nine months with Fraleigh and Ms. Chen. Yes, the subject is challenging, but far from insurmountable.If you have a less than enthusiastic and absent-minded professor that seems indifferent, do not let this be an obstacle to your learning either.Go buy these other books and use Fraleigh as a supplemental.Be consistent in setting time aside to read and do the problems of the text.Work as independently as possible and then go to group meetings-not the other way around, as is often told by instructors.I'm now a third year graduate student and focusing my work in algebraic topology-of all things!And I love it.This subject is a real gem and will doubtless never cease to impress you as you grow with it.Best wishes in your learning and educational endeavors.

Classic textbook
THE definitive text for introductory algebra classes, this book is a classic. And for good reason. It introduces each concept with solid examples and a lengthy discussion. There are always a few nice problems for each chapter, some of which provide meaningful results for later sections. One criticism of this book is that Fraleigh often assumes that the reader has completed all of the problems and forgoes explanation on many areas. Overall, he presents the material in a nice, clean package: perfect for a first course in abstract algebra.

Reviews (4)

good book
This book (5th edition) cover the topics of undergraduate number theory well. The chapters are -
(1)divisibility
(2)congruences
(3)quadratic reciprocity and quadratic forms
(4)some funtions of number theory
(5)some diophantine equations
(6)farey fractions and irrational numbers
(7)simple continued fractions
(8)prime estimates and multiplicative number theory
(9)algebraic numbers
(10)partition funtion
(11)density of sequences of integers.
It also contains basic cryptography, basic group theory and basic elliptical curves in some of the chapters. The authors give notes on the end of each chapter about some research results, which I enjoy reading.

However, the author give too much hints spoling the fun of solving the problems. Eg 32-36, 40-3, 59-53, 108-36, 136-17, 312-8, and most of the problems in chapter 8. The author should put these hints at the back of the book. I suggest you look up IMO (imo.math.ca) for problems suitable for chapter 1-7 because IMO is well-knowned for its excellent number theory problems (especially 1990-3).

Overall this is an excellent book. I give it a rating of 4.5/5, I don't give it 5 because of the author give too much hints to problems instead of putting hints at back of the book.

Comprehensive
This is a fantastic book on number theory. It covers far more ground than most introductory text (comparable to Hardy and Wright in depth with much less concern for the big O). It covers material usually only available in separate texts: Rational points on elliptic curves, the partition function, and Dirchlet series.Quite readable chapters, well motivated theoretically, although the historic motivation for the subject matter comes largely in the end-of-the-chapter notes.It's an excellent refresher and reference for non-specialist who find themselves using an algorithm or formula they've forgotten(number theory now playing a role in physics and CS, like never before). It is well cross-referenced with regards to methods of proofs the can be accomplished in different section by different methods - this again making it an excellent reference.

Alas, it is pre-FLT. So you'll have to look elsewhere for that.

Reviews (1)

Reviews (18)

FRUSTRATING
First of all, a little background on myself. I am a college sophomore and an overachiever in mathematics for my entire life. I am currently taking a Linear Algebra course that uses this book. We have just finished chapter 2 and I am ready to throw my book in the fire.

Before I talk about why I hate this book I will first give some positives.

1. The book is very concise. A lot of material is covered in relatively few pages. There are not many pictures or graphs to clutter up the pages.
2. This book covers the material very thoroughly. Everything is defined and everything has a clear proof, except for a few theorems which are left as exercises.

Now for the bad...

1. The thing that drives me insane about this book is the lack of decent examples. Since this is new material to myself and many others, it really helps to aid understanding if there are well-explained examples to accompany the new information. Most of the examples in the text are either WAY too simple, or are left thoroughly unexplained. Sometimes I feel like a toddler trying to learn addition, but all I have to go on is that 0+1=1.

2. There is no answer key. If there were an answer key for the problems it would more than make up for the lack of examples, but instead we are left with about 5% of the answers to the simplest exercises.

3. There is little explanation of the computational aspects of linear algebra. This goes hand-in-hand with the poor examples. If the text spent as much time explaining how to use the material as it does explaining where it is derived from then it would be a lot easier to understand.

That's all I can think of for now. This is definitely not the worst textbook in the world, but my opinion is that the bad outweighs the good and it should be avoided by anyone new to Linear Algebra.

Hope this helps.

An Extremely Frustrating Book
I have never seen a book which hides so many important details (and even basic definitions) in exercises. This book was terrible to learn from and even worse as a reference. It also omits or glosses over a number of important topics including quadratic forms and matrix norms.

Reviews (12)

A Bible for Linear Algebra
This is a holy bible for linear algebra beside P R Halmos(Finite dimensional vector spaces). I can assure you that if you understand this book, no one can teach you linear algebra no matter you go. Personally, I will rate this book 7 stars and 7 diamond. It is a superb. Buy and don't miss it. Add to your collection.

Best intro to linear algebra I've seen.
I simply loved this book.Hoffman and Kunze have written a very sturdy book that begins with the most basic concepts of linear algebra(such as echelon form) and goes through cannonical forms, inner product spaces, and Bilinear forms.The proofs are complete and at an appropriate level for a first look at the subject.Perhaps one of my favorite aspects of this book was the treatment of dual spaces and tensors.It seems many linear texts deal with one subject or the other but rarely do I see both subjects dealt with in the same book.

The only non-positive comment I would like to make about this book is that its beauty is not in its appearence.When you open the book and flip through the pages you feel a little uneasy.The typeset looks uninviting.Take heart!The beauty of the book lies in its content.Give it a thorough chance, and I don't think you will be disapointed

I highly recomend this book for both learning and reference.

Reviews (73)

An excellent textbook
I think mathematics is a part of our culture.That's why, as a non-math major, I wandered into a very serious analysis class for mathematics majors.That might have been a disaster for me.Luckily, we used this book as a text, and it saved me.I read the whole book and diligently did all the exercises (of course, back then, it was the first edition, with only 227 pages and 140 exercises; it's somewhat more now).And that is my recommendation today.Read the book carefully and do as many exercises as you can.It certainly isn't easy.But it isn't, um, countably hard either.

The material in the book is self-contained.I guess that in theory, it could be mastered by any bright 14-year old who had learned some high school algebra and geometry.But I would surely recommend having much more mathematical sophistication than that as a prerequisite!

If you haven't learned the language of mathematics before, you'll enjoy the use of terms such as "countable," "real," "rational cuts," "measure," "ring," and "complete." By the end of the book, when the author claims that a proof (involving Cauchy sequences no less) is complete, you'll barely be able to suppress a desire to ask "Does every Cauchy sequence in the proof converge?"

In the first edition of this book, Rudin did mess up a little in his section on "the integral as a limit of sums." His theorem as stated was false.We cruelly dubbed it "Rudin's Last Theorem."Worse, he had used it "to prove some elementary properties of the Stieltjes integral."But that was all straightened out by the second edition.

I especially like the first couple of chapters.They give most readers the confidence to continue.And the final chapter, on Lebesgue integration, is very well written.One note of warning, though.Rudin begins this chapter by saying, "Some of the easier propositions are stated without proof.However the reader who has become familiar with the techniques used in the preceding chapters will certainly find no difficulty in supplying the missing steps."That is an exaggeration.It takes work.After all, this is, um, real mathematics you'll be doing!

I'm thankful that I was assigned this as my textbook.

A masterpiece
I absolutely agree with Professor Jorgensen.

I loved it when I was a student of physics, and I still love it because I tend to consider it as my personal standard in Classical Mathematical Analysis (and not only): sort of a "pacemaker" which sets the qualitative level to go back to just every time one is a little confused about what to do and where to go ;)

Reviews (24)

The best rigorous introduction to general topology!
I have owned the 1975's first edition (red cover) of this book which I am currently studying again to pass a Ph.D. qualifying exam on topology. From the many topology texts that I have come across over the years, this one easily stands out as the best rigorous introduction to point set topology for a beginning graduate student. It covers all the standard material for a first course in general topology beginning with a chapter on set theory, and now in the second edition includes a rather extensive treatment of the elemantary algebraic topology. The style of writing is student-friendly, the topics are nicely motivated, (counter-)examples are given where they are needed, many diagrams provided, the chapter exercisesrelevant with the correct degree of difficulty, and there are virtually no typos.

The 2nd edition fine tunes the exposition throughout, including a better paragraph formatting of the material and also greatly expands on the treatment of algebraic topology, making up for 14 total chapters (as opposed to 8 in the first edition). A notable minor issue in the first edition was the consistent usage of the pronoun "he" in the discussions for addressing all the possible readers of the book. (This fortunately has been modified in the 2000's edition.) On another note, I wish there were some hints & answers provided at the back of the book to some of the harder problems, so as to make this text more helpful for those of us who use it for self-study.

One of the two spotlight reviewers has correctly mentioned that Munkres does not cover differential topology here. I speculate this is perhaps because Munkres has already a separate monograph on differential topology. It is also necessary to get a handle on some fair amount of algebraic topology first, for a full-fledged coverage of the differential treatment. Regardless, one great reference for a rigorous and worthwhile excursion into differnetial topology (covering also Morse Theory) is the excellent monograph by Morris W. Hirsch, which is available on the Springer-Verlag GTM series.

At the end, I shall mention that one other very decent book on general topology which has unfortunately been out of print for quite some time is a treatise by "James Dugundji" (Prentice Hall, 1965). The latter would nicely complement Munkres (for example, Dugundji discusses ultrafilters and some more of the analytic directions of the subject.) It's a real pity that the Dover publications for example, has not yet published Dugundji in the form of one of their paperbacks.

great!
Not much to add here... there are enough easy problems that I can get the hang of something, but also some really tough ones at the end of each problem section. The proofs and examples in the text are really good guides to doing the problems also. In some sections there are counterexamples for, say, the converse of a theorem which are always really pathological. At the beginning of each section there is some discussion on what to expect, why the stuff is important, what to do with it, etc. Even though I had a really good prof for the topology course I did this book was very helpful out of the classroom.

Excellent Topology Book
My introduction to Munkres was in an independent study of point set topology in my final semester of undergraduate work. A professor assigned me problems from the book, but my learning was largely self motivated.I found that it was an excellent book for independent study.The text was clear and readable and the exercises helped to cement the concepts that are introduced in the reading.

Later at graduate school, Munkres was also used in a topology class at the beginning graduate level.Highlights were taken from the first section (point set topology), and a large focus of the class was on the algebraic topology in the second section of the book.Sometimes I had difficulty following exactly what the professor was doing at the blackboard, but I could always understand what was going on when I consulted Munkres.

Reviews (3)

Unsurpassed SECOND text on number theory
The amazing positives of this book are accurately described in the other reviews so I will skip them.There is no negative, but the other reviewers assert that the reader needs no prior exposure to number theory.I completely disagree.

While this book does quickly cover elementary number theory, a reader new to this field will quickly feel lost.Without more exposure and a good prior feel for elementary number theory, the use of analytic techniques will seem ad hoc instead of following a logical pattern.By way of example, three areas covered in this book that are not part of analytic number theory and for which the reader would do better to learn from a less sophisticated text are the Fermat-Euler Theorem, Diophantine equations, and quadratic reciprocity.

Excellent texts for a first exposure to number theory are, from simpler to more difficult:

1. Elementary Number Theory by Underwood Dudley

2. An Introduction to the Theory of Numbers by Niven, Zuckerman and Montgomery

3. An Introduction to the Theory of Numbers by Hardy and Wright

Apostol's book on analytic number theory is a classic that may never be surpassed.It is a marvelous second book on number theory.

well presented, delightfully written
I think that there will be little harm if the title of the book is changed to 'Introduction to elementary number theory' instead. The author presumes that the reader has not any knowledge of number theory. As a result, materials like congruence equation, primitive roots, and quadratic reciprocity are included.
Of course as the title indicates, the book focusses more on the analytic aspect. The first 2 chapters are on arithmetic functions, asymptotic formulas for averaging sums, using elementary methods like Euler-Maclaurin formula .This lay down the foundation for further discussion in later chapters, where complex analysis is involved in the investigation. Then the author explain congruence in chapter 4 and 5. Chapter 6 introduce the important concept of character. Since the purpose of this chapter is to prepare for the proof of Dirichlet's theorem and introduction of Gauss sums, the character theory is developed just to the point which is all that's needed. ( i.e. the orthogonal relation). Chapter 7 culminates on the elementary proof on Dirichlet's theorem on primes in arithmetic progression. The proof still uses L-function of course, but the estimates, like the non-vanishing of L(1) , are completely elementary and is based only on the first 2 chapters.
The author then introduce primitve roots to further the theory of Dirichlet characters. Gauss sums can then be introduced. 2 proofs of quadratic reciprocity using Gauss sums are offered. The complete analytic proof, using contour integration to evaluate explicitly the quadratic Gauss sums, is a marvellous illustration of how truth about integers can be obtained by crossing into the complex domains.
The book then turns in to the analyic aspect. General Dirichlet series, followed by the Riemann zeta function, L function ,are introduced. It's shown that the L- functions have meremorphic continuation to the whole complex plane by establishing the functional equation L(s)= elementary factor * L(1-s). The reader should be familiar with residue calculus to read this part.
Chapter 13 may be a high point of this book, where the Prime Number Theorem is proved. Arguably, it's the Prime Number Theorem which stimulate much of the theory of complex analysis and analyic number theory. As Riemann first pointed out, the Prime Number Theorem can be proved by expressing the prime counting function as a contour integral of the Riemann zeta function, then estimate the various contours. The proof given in this book , although not exactly that envisaged by Riemann , is a variant that run quite smoothly. As is well known , a key point is that one can move the contour to the line Re(s)=1, and to do this one have to verify that zeta(s) does not vanish on
Re(s)=1.The proof , due to de la vale-Poussin, is a clever application of a trigonometric identity. Unfortunately, the method does not allow one penetrate into the region 0The last Chapter is of quite differnt flavour, the so-called additive number theory. Here the author only focusses on the simplest partition function ---the unrestricted partition. However interesting phenomeon occur already at this level. The first result is Euler's pentagonal number theorem, which leads to a simple recursion formula for the partition function p(n). 3 proofs are given. The most beautiful one is no doubt a combinatorial proof due to Franklin. The third proof is through establishing the Jacobi triple product identity, which leads to lots of identites besides Euler's pentagonal number theorem. Jacobi's original proof uses his theory of theta functions, but it turns out that power series manipulaion is all that's needed.
The book ends with an indication of deeper aspect of partition theory--- Ramanujan's remarkable congrence and identities ( the simplest one being p(5m+4)=0(mod 5) ). To prove these mysterious identites, the "natural"way is to plow through the theory of modular functions, which Ramanujan had left lots more theorem ( unfortunately most without proof). However an elementary proof of one these identites is outlined in the exercises.
This book is well written, with enough exercises to balance the main text. Not bad for just an 'introduction'.

Excellent exercises in a clear exposition
This book has excellent exercises at the end of each chapter. The exercises are interesting and challenging and supplement the main text by showing additional consequences and alternate approaches.