This is a Dover reprint of a text, first published in 1983, covering the basics of point-set and algebraic topology at a fairly sophisticated undergraduate level. Unlike many Dover reprints, which are unaltered, this one differs from the original in that it contains an appendix, almost 40 pages long, of solutions to most of the exercises.
The text is divided into four chapters, each subdivided into numerous sections. The first two chapters are on metric and topological spaces, respectively; the third and fourth treat, first, homotopy theory (at about the level it is treated in Munkres’ Topology) and then — an unusual feature of books at this level — higher homotopy groups and related topics.
The chapter on metric spaces is quite complete and contains a mini-course in topology in the context of these spaces, with an emphasis on applications to analysis. Continuity, compactness and connectedness, and product spaces are introduced and discussed, first for metric spaces and then, in the next chapter, in the more general setting of topological spaces. (This approach, which I think is the optimal one for learning these ideas, is also the approach used by Conway in his A Course in Point-Set Topology, but Conway’s book sticks to point-set theory and doesn’t get involved with the fundamental group.)
The metric space chapter in the book under review also pursues some ideas beyond that which is usually done in undergraduate texts at this level. For example, while it is not uncommon for books to mention normed vector spaces as examples of metric spaces, this book takes matters further and goes into topics that are often associated with courses in functional analysis, such as the definition of a Banach space and a statement and proof of the Uniform Boundedness Theorem. Complete metric spaces are introduced, which is again pretty standard, but here the authors go so far as to state and prove the contraction mapping theorem, and give examples to analysis (integral equations, differential equations). This is by no means the only time that quite nontrivial analysis is discussed in the text; Frechet derivatives, for example, are also the subject of a section in the first chapter, and proofs of the implicit and inverse function theorems (in the context of Banach spaces) are given.
In the chapter on topological spaces, the authors not only revisit topics previously introduced but also introduce new ones, such as the separation axioms, path-connectedness, and infinite product spaces (including a proof of Tychonoff’s theorem for general products). To prepare the way for this latter topic, there is a section on set theory, including Zorn’s Lemma; here again the discussion goes further than one often sees, and includes, for example, a proof of the existence of bases for arbitrary vector spaces.
Chapter 3 on homotopy theory discusses the fundamental group, covering spaces, and related ideas. The concept of the index of a mapping is introduced and then used to prove some standard applications, including the Borsuk-Ulam theorem and the “ham sandwich theorem”. Vector fields and the famous “hairy ball” theorem are discussed, and the Jordan curve theorem is stated and proved.
The next chapter introduces the higher homotopy groups \(\pi_n(X)\). These are defined, not in terms of loop spaces and the compact-open topology, but in terms of equivalence classes of mappings from the n-fold Cartesian product of \([0,1]\) with itself; the connection between these definitions is mentioned briefly but not elaborated on. Other topics discussed in this final chapter of the text are the non-contractibility of the n-sphere \(S^n\), the Brouwer fixed point theorem, simplices and barycentric subdivision, and the uniform approximation of continuous maps from a simplex to a Euclidean space by piecewise affine maps. A final section uses the concept of degree to prove that \(\pi_n(S^n)\) is infinite cyclic, and ends with a brief look at the concept of a manifold. This final chapter strikes me as being a bit too sophisticated for the average undergraduate.
The writing throughout is quite elegant but very concise, perhaps too much so for the current generation of undergraduates. The first two chapters, for example, which probably contain enough material for most of a one-semester course, comprise only about 100 pages. There is also a paucity of examples in the body of the text; in most books, for example, the page or two immediately following the definition of the term “topological space” would consist of a list of examples, but here there is only brief mention of the discrete and indiscrete topologies, followed by a statement that any metric space is a topological space. (The cofinite topology is mentioned in the exercises.)
And speaking of the exercises: they appear at the end of each section, are fairly numerous, and range from the routine to the quite challenging; the famous Kuratowski “14 problem” appears as one, for example (with a nice hint). Some exercises extend the theory developed in the text by calling for proofs of very well-known results; a couple, for example, discuss equicontinuity and the Arzelà-Ascoli theorem, and another introduces the topologist’s sine curve as the standard counterexample to the converse of the theorem that path-connectedness implies connectedness. As mentioned earlier, this Dover edition contains solutions at the end of the book; these range from fairly complete to those that are as concise as the text itself (e.g., “follow the hint”).
In my earlier review in this column of Munkres’ book Topology, I compared that book to Rudin’s venerable Principles of Mathematical Analysis (aka “Baby Rudin”); having now seen this book, though, I realize that it, rather than Munkres, is the true analog of Rudin. That’s hardly a criticism, of course; many of us who learned analysis from baby Rudin have great affection for that beautifully written book, even if we now recognize that it could not effectively be used as a text for an introductory course at an average university. That, I think, summarizes my feelings about the book under review as well — a very nicely written, concise account of the material that certainly belongs on the shelf of anybody interested in topology (at its current price of about ten dollars on amazon.com, how could any such person pass it up?) but probably not the book to select as a text for a first course on the subject.
Mark Hunacek (firstname.lastname@example.org) teaches mathematics at Iowa State University.