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 (1)

Reviews (1)

Excellent for first-year graduate students
Emphasizing geometric topology, the authors have written an excellent introduction that is suitable for students of topology at the beginning graduate level. It would take two semesters to cover all of the material in the book, and there are lots of exercises and problems to challenge the reader's understanding. The main virtue of the book is that the authors do not hesitate to use pictures and diagrams to illustrate the concepts in topology.

The first two chapters are an overview of elementary set theory andbeginning notions in topology, such as metric spaces, the product topology, etc. The authors assume that the reader has encountered these ideas before, and so they do not spend a lot of time explaining them. Calling one section a "potpourri of fundamental concepts" the authors define accumulation points, closure, etc. There are some interesting insights though that the authors give the reader, particularly in the discussion on homeomorphisms. They caution the reader, giving a pictorial example, that thinking of a homeomorphism as a stretching or deformation is a somewhat limited view. The example shows two objects that are homeomorphic but that cannot be deformed into each other in Euclidean 3-dimensional space.

The next two chapters cover connectedness, compactness, and metric spaces. The authors show what pathologies can arise in the consideration of connectedness, challenging the reader to find an example of a space with an explosion point. They also define the notion of a chain, a concept that proves to be very useful in geometric topology. The authors motivate its eventual application very well, their construction beginning with an arbitrary "entangled" collection of open sets, out of which the chain is systematically selected. The famous Knaster-Kuratowski example is discussed. For readers interested in moving on to dimension theory, this example is important, in that it is a one dimensional set that is not totally disconnected. Separation properties are discussed in Chapter 4, and again reflecting their prejudice for geometric topology, the authors define and discuss absolute retracts and absolute neighborhood retracts.

Things get very geometric in chapter 5, wherein topology of the Euclidean plane is discussed. The Jordan curve theorem is proved in detail, along with the Schoenflies theorem. The latter has to rank as one of the more amusing results in geometric topology, and its proof is a joy to construct. Then, in chapter 6, the authors return to the consideration of product spaces, and they also define and discuss inverse systems. An understanding of inverse systems is a must for readers intending to move on to algebraic topology. The dyadic solenoid, an important construction in the field of dynamical systems, is discussed geometrically and then shown to arise as an inverse limit.

Considerations of a more analytic nature appear in chapter 7, which deals with function spaces, weak topologies, and Hilbert spaces. The compact-open topology, important in many area of application, is discussed as a topology that guarantees that a sequence of continuous functions converges to a continuous limit. The weak topology is introduced as a generalization of the free union topology, and its importance in the study of cell complexes is pointed out.

The glueing and identification operations, so familiar from popular or more elementary expositions of topology, are discussed in chapter 8. These are the quotient spaces, and the authors discuss the cone and suspension of a space as examples. CW-complexes are then introduced and discussed in detail. This is followed in chapter 9 by a discussion of one of the most important of all topological spaces: continua. The Peano continua in the light of the Hahn-Mazurkiewicz are overviewed.

If the reader has studied differential geometry, then chapter 10 will be somewhat familiar, as it deals with paracompactness and partitions of unity, the later of which are used extensively to perform some very standard constructions in the theory of differentiable manifolds. Metrizability is also discussed, and the authors give an example of a Moore space that is not metrizable.

Chapter 11 gives an alternative view of convergence, wherein the authors discuss nets and filters. The pathologies that can arise for sequences in non-metric spaces are emphasized. Filters may be familiar for the reader who has studied mathematical logic, where they are used extensively.

Things heat up in chapter 12, wherein readers get to indulge in the intricacies of algebraic topology, a topic that has been hinted at in a few places in the first eleven chapters. Homotopy theory and the fundamental group make their appearance, as well as the notion of a direct limit. The higher homotopy groups are introduced in the problem sets. The reader versed in algebra will certainly appreciate this chapter, as well as the next one, which deals with covering spaces, which the authors mention is a topological analog of Galois coverings. Covering spaces allow the computation of the fundamental group, as well as being useful in many other applications.

Simplicial topology is introduced in chapter 14 as objects that have a local linear structure, and can thus be studied much more easily than more general typesof spaces. Most readers will catch on very quickly to this category of spaces, due to its connection with notions from plane and solid geometry, and linear algebra. The simplicial approximation of maps is emphasized, with an elementary example of a continuous map that cannot be simplicially approximated given. A hint of the field of simple homotopy theory is given in the problem section, with the famous Bing's house with two rooms discussed.

Reviews (1)

Reviews (9)

Deceptively Wonderful
OK here's the truth:This book is an awful text when accompanied by not so great prof is teaching from it (e.g. one who delivers nothing but the text). BUT... once you begin to understand enough to know that the "trivial" "exercise" and "left to the reader" proofs are quite straightforward, the book is probably the best reference in Algebra you can hope for.

Well worth the read for any budding mathematician
I've been acquainted with several introductory graduate algebra
books over the years, and prefer this one for its coverage of all the fundamental areas (groups, modules, rings, linear algebra, fields, and category theory), being concise, and providing great care when outlining each proof.

If one compare's the amount of material in this book to Jacobson's "Basic Algebra Vol 1", Grove's "Algebra", or Herstein's "Abstract Algebra", Hungerford's book gets the nod.
Moreover, I much more prefer the concise definition, example, theorem, proof format over the more colloquial approach, as can be found in Jacobson's text. For me at least, the payoff for reading an algebra text is the beauty found in the logic and reasoning from which very profound results arise from the complex interaction and use of more straightforward ones. And this is exactly where Hungerford's book shines through in tremendous glory. When outlining a proof he does an outstanding job in citating the results from previous Chapters that are used. For me this is the strength of algebra (In geometry I cringe when I get a picture for proof, and in analysis it is often quite complicated to verify that a given situation possesses the appropriate conditions needed to invoke some famous lemma or theorem).

One last good word about this book: I found the exercises both in abundance (after each section) and quite reasonable for a first year grad. student. Happy reading.

The bee's knees
This book is the Basic Language of Mathematics (by J. J. Schaffer) of the Algebra world.Without doubt it is an excellent dictionary of general facts about algebra.But learning by it will leave one with at best amusing memories and a nervous twitch.Just for a taste, "This proof has two parts.The first is easy.The second is left to the reader."About half the proofs in the book go like this.And so at the end of each section, the reader is left with just the dry theorems to attempt the exercises, without the slightest idea of how problems of a certain type are actually proven or even approached.And oh, the exercises.A few are easy.A few are open problems.The rest in between seem to at one point have been at the core of someone's respective masters thesis.

This book has three genuinely good uses.If you have a doctorate in pure Mathematics, a respectable doctorate that has nothing to do with PDEs and the thesis for which took longer to write on paper then it did to format the pictures to fit the margins, and you want to look up how much of the ring structure of R is inherited by R[x] in under 3 minutes, then this book belongs on your shelf.

If you have taken at least two algebra courses at the graduate level (Real graduate, not graduate equivalent.Most of my Algebra I class had two pretty good undergrad algebra classes coming in, and got slaughtered by Hungerford), then this book can make for a good review of basic algebra you should already know.

Reviews (4)

no better than the first edition
It is always easy to add something to than to get rid of something from the book. I guess this is the case of the author when he prepares the second edition. However, I prefer the first edition because it is more readable, enjoyable, and most importantly, contains just enough information for the introduction to abstract algebra. There are huge number of textbooks on abstract algbra, and making another would not be the author's purpose of the revision, I hope, but it looks it is.
By adding more subjects in detail to the second edition, now it looks the same as any other, only to loose its expository and conversational style of writings, and became a reference-style textbook.

Boo
Before taking an abstract algebra course this semester I studied the material on my own using the introductory texts by Gallian and Hungerford.These books were very useful because they actually completed proofs instead of leaving them as exercises for the reader.Someone new to abstract algebra is also typically new to higher mathematics.This means a book should have clear and full explanations, not skip major points like Rotman does.Rotman commits another sin by failing to provide homework problems which correspond with the material he presents. One nice thing is that the book does provide a wide array of material (much more than most other introductory texts).This virtue soon turns astray however because by providing so much preliminary material on congruences, functions, divisibility, .... you'll be lucky if your teacher gets to groups by halfway through the semester.

Reviews (4)

The worst mathematics book I have ever read!!!
I gave this book one star only because I couldn't give it a score of zero!!! Although many professors say that this book is excellent, remember they are professors who already understand the material. This book shows no examples, and the examples that it does show end abruptly with comments such as "all items are routine." Routine!!! Please show me what to do so that I don't have to spend more money on a separate study guide. Aren't mathematics texts expensive enough? This book may be an excellent addition to a professors library but this book should never, ever be used as a primary text for students.

Good for Self-Study
This is a tough book to review, because it is not clear who the real audience is supposed to be.The author says that it is aimed at first-year graduate students, with a bunch of extra material that can be referred back to during the second year and beyond.The earlier chapters also include efficient reviews (with sketched proofs) of material that should be familiar to those who have taken undergraduate algebra.

This characterization is debatable.Based on my experience reading most of the first six chapters (the first 400 out of about 1000 pages), I would say that the level of sophistication is roughly that of Dummit and Foote's "Abstract Algebra", which is usually considered an undergraduate book.D&F can sometimes be harder to read, and that is in part because Rotman's exposition is better (in my opinion), but also because D&F introduce more difficult material earlier.Whether D&F's approach is better is questionable; I find Rotman to be a much smoother read, but the organization is quite different -- for example, one does not encounter noncommutative rings until deep into the book, whereas Dummit and Foote introduce them immediately upon defining rings.On the other hand, early in the coverage of D&F's chapter on rings, one has to digest Zorn's Lemma and its applications almost from the beginning, whereas Rotman (I think wisely) pushes this back into a later section.In general, D&F introduce a lot of hairy examples that by themselves require a lot of effort to digest (thereby impeding the reader's progress through the core material), whereas Rotman's examples tend to be straightforward, at least as new concepts are being presented.

So, overall, the exposition flows more smoothly in Rotman's book, and the reader can cover the basics more quickly with less time spent on tangential examples and early generalizations.Also, Rotman's proofs are usually much cleaner and the overall style is very nice.It's more pleasant to read than Dummit and Foote.But this comes at a cost: Dummit and Foote do cover more material, and generalize at an earlier stage, than Rotman does.

But my biggest gripe concerns the exercises.Put simply, Rotman's are far too easy for what is being pitched as a graduate course.In fact, they are in general far easier than the homework problems I sweated through when I took honors undergraduate algebra. They're barely adequate to convince the reader that he has a basic grasp on the material, and there are almost no hard ones, let alone really tough, thought-provoking open-ended problems like one encounters in Herstein's "Topics in Algebra" (an undergraduate book).There are certainly no exercises in Rotman's book that would be of any use for a graduate student preparing for qualifying exams.They're not even much of a workout for a decent (honors student) undergraduate.

So, what is this book good for?I think it's great for reading material that is usually harder to understand elsewhere.Rotman has a real knack for clear mathematical exposition, and some of the chapters are a real joy to read.(Side note: there are also a lot of typos, at least in the first printing.The author maintains an errata list at his web site, and a second printing is coming soon.There are still many errata that he didn't catch, but they're fairly minor and do not detract significantly from the reading.) But this is simply not suitable for a primary graduate text or reference.Most good schools are going to demand more of their graduate students, and one is inevitably going to have to read Lang or Hungerford (and work through their exercises) to achieve competence at the graduate level.Rotman's book is a kinder, gentler book upon which to fall back when those books are inscrutable, as is all too common.I do recommend it highly for that purpose -- I think it's a very good secondary book.

An excellent Text
To begin with, don't let the title scare you.After having read through Rotman's book I am suprised that this text had not crossed my path earlier.It is a wonderful book and must have for any inspiring Algebraist.Moreover, I am quite shocked that the larger universities have not adopted this book.

(a)This book could quite easily be used as the standard third/fourth year undergraduate introduction to Abstract Algebra.In particular, the first four chapters provide a solid foundation for a moderate paced one semester course at which point the instructor has many different options for additional topics based on the performance of his/her class.

(b)Those students that move on to the graduate level, and obviously to a university using this book, would both be familiar with the temperment and flow of the author as well as devoid of the requirement of having to purchase another expensive Mathematics text.For example, my undergraduate Algebra text was Hungerford's and post completion the logical step, being familiar with his style, was to purchase Hungerford's graduate text.For those not familiar, let me tell you there is a night and day difference with repsect to how the material is presented.

(c) The remaining 7 chapters take the willing student on a pleasant tour of ring/module theory, some advanced group theory (for the inspiring group theorist I highly recommend the authors graduate text "Group Theory"), algebras(linear included), Homology(some cohomology) and finally some algebraic number theoretic concept under the heading of Commutative Rings III.

(d) Lastly, Rotamn does not get needlessly bogged down in any one section of the book.The flow is smooth, to the point with precise definitions, examples, and ample exercises.

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