This venerable book has been around, in one form or another, for almost 40 years now. This is not a coincidence or an accident: *Topology* is an exceptional book, and, though I have some doubts as to its current suitability as an undergraduate text at an average institution, it certainly is a book that should be owned by anybody who teaches such a course or who has an interest in the material. It is a genuine classic.

The book was originally published in 1975 under the title *Topology: A First Course*, and was described in the preface as a text for topology at the senior or first-year graduate level. Of course, one had to bear in mind that the author taught at MIT, so what was suitable for undergraduates there might not be suitable for undergraduates at other places. The book covered point-set topology quite thoroughly, and also contained a brief introduction to algebraic topology in the form of the fundamental group. I have considerable fondness for this first edition, since, at just about that time, I was taking a graduate course in algebraic topology with a teacher who, shall we say, gave whole new meaning to the word “incomprehensible”; I purchased the book and thought it was the clearest introduction to the fundamental group that I could find in print.

Twenty-five years later, in 2000, the second (and current) edition appeared, this time titled simply *Topology*. This is a substantial revision, with a greatly expanded introduction to algebraic topology at the level of fundamental groups, covering spaces and surfaces; the deletion of the phrase “First Course” from the title undoubtedly reflects the fact that the book now contains enough material for both a first and second course. This division is explicitly reflected in the text, which is divided into two parts, “General Topology” and “Algebraic Topology”, containing eight and six chapters, respectively.

Part I on general topology opens with a chapter on set theory and logic that is much more detailed and comprehensive than one typically finds in books at this level. It is more than 70 pages long, and goes into some depth on things like the Axiom of Choice, Maximum Principle and well-ordering (the order topology is used throughout the book, later). The next chapter begins the subject of topology with the definition of a topological space. From a strictly pedagogical point of view I think there is some value in first discussing metric spaces in detail, thus motivating the next level of abstraction to topological spaces, but Munkres defers metric spaces until later in the chapter, where they appear as examples of topological spaces. This different point of view may reflect a fact that I alluded to in the first paragraph above, and which I’ll discuss in more detail below: what was suitable for undergraduates 40 years ago may not be suitable today.

The remaining chapters in part I of the text cover most of the standard topics in point-set topology (compactness, connectedness, separation axioms, normal spaces, Tychonoff’s theorem, Baire category), as well as a number of topics that might not be covered in an introductory course (e.g., Nagata-Smirnov metrization theorem, paracompactness, compact-open topology, dimension theory). Several additional topics (e.g., topological groups and nets) that are not covered in the main body of the text are developed through a sequence of supplementary exercises.

Part II of the book is a beautiful introduction to algebraic topology. Topics covered here are the fundamental group, covering spaces, and the classification of surfaces (including a brief look at the first homology group, defined as the abelianization of the fundamental group rather than the usual way). Along the way we encounter, for example, a proof of the Fundamental Theorem of Algebra using the fact that the fundamental group of a circle is infinite cyclic, and also the Borsuk-Ulam theorem. There are also chapters on the Seifert-van Kampen theorem, the Jordan curve theorem and other separation theorems in the plane, and a theorem in group theory (that every subgroup of a free group is free) that can be strikingly proved via algebraic topology.

This part of the text is comparable, in both content and level of difficulty, to Massey’s excellent *Algebraic Topology: An Introduction*, though of course Munkres also has the advantage of also containing a lot of general topology material in part I, thereby making this book more suitable for instructors offering a two-semester course combining both areas. Massey’s book is clearly intended for a graduate-student audience (it appears in Springer’s Graduate Texts in Mathematics series) and so, I think, is Part II of Munkres; this is material that is just not covered very much at the undergraduate level.

Of course, many universities do offer undergraduate point-set topology courses; is this book suitable for one of those? While this text is certainly being used at undergraduate institutions across the country, I would have some trepidation about using it for our undergraduate course at Iowa State. This is a text that (in Part I as well as in Part II) both requires and rewards serious reading by a well-motivated student. Explanations are clear, but concise, and there is not a lot of hand-holding. I have already pointed out, for example, that the definition of “topological space” comes without much in the way of prior motivation; this is typical of the no-nonsense, direct approach to the subject that characterizes the text. The writing is elegant and the student is provided everything he or she needs to learn the material well — there are lots of pictures, and many exercises (some relatively easy, but quite a few somewhat challenging) — but the student is expected to put in some hard work reading the book. The very best students in the class could probably get a lot out of reading it, but the more average students might find it daunting and discouraging.

I would say, by way of rough analogy, that this book is to undergraduate topology what Rudin’s *Principles of Mathematical Analysis* (aka “Baby Rudin”) is to undergraduate analysis: I used Rudin as a college student, loved it, and learned a tremendous amount from it, but if I used it as a text for the undergraduate analysis course here at Iowa State, the results would not be pretty.

Having said this, I hasten to point out that I would not want to see a new, dumbed-down, edition published of either Rudin or Munkres. The originals have a kind of elegance and nostalgic charm to them, a reminder of what college mathematics courses were once like. Even if its use as a text might be problematic, this book is still the one I look at first when I have a question about point-set topology. I see that the Basic Library List Committee has given this book three stars, meaning that the Committee views it as essential for undergraduate libraries; I couldn’t agree more.

Mark Hunacek ([email protected]) teaches mathematics at Iowa State University.