Introduction to Topology

This is a book that sends mixed signals. The front cover proclaims it to be part of a graduate series, but the back cover says its aim “is to give a broad introduction of [sic] topology to undergraduate students.” As the reviewer, I get to be the referee: my verdict is that this book is undergraduate-level. It starts pretty much from scratch with the very basic concepts of sets, functions and relations, proceeds through standard undergraduate material in topology, and, except for one annoying feature noted later (and hinted at in the above quote), is quite successful.

The book covers, in more or less alternating chapters, the rudiments of point-set and combinatorial topology. The point-set theory (chapters 1, 4, 5, 7, 9) begins with metric spaces (a good choice; I have always felt that starting with metric spaces helps motivate the more abstract definition of a topological space) and then proceeds to topological spaces, covering such familiar topics as product and quotient spaces, connectedness and path-connectedness, and compactness. The approach is generally elementary, avoiding any of the really difficult results until the final chapter on advanced topics, which is a step up in difficulty from the ones preceding it. In this chapter, for example, the author explains how the concept of a product space (previously defined only for finite products) can be generalized to arbitrary products, and proves Tychonoff’s theorem. It is also only in this final chapter that complete metric spaces are defined and the Baire category theorem proved (as an application of the latter, there is a proof that there exist functions that are continuous everywhere but differentiable nowhere). Other topics covered here include normal and paracompact spaces, space-filling curves and spaces of mappings.

Interestingly, bases are introduced even before topological spaces are: the author defines a topological basis as a collection of sets satisfying certain conditions (giving as an example, of course, the set of open balls in a metric space), and then uses this definition to define open sets, which in turn leads to the definition of a topological space. Most textbooks at this level that I have seen define topological spaces and then bases for a topological space, but the author’s approach is certainly a credible one.

The chapters on combinatorial topology (2, 3, 6 and 8) discuss graphs, surfaces, CW complexes and manifolds. Topics covered here include Euler’s formula, the Euler number of a CW Complex, and the classification theorem of surfaces.

There are several things about this book that I liked. For one thing, it seems to have been written with the needs of a beginning student in mind. Explanations are generally detailed and clear, and are accompanied by many pictures (though I wish some had been larger) and well-chosen illustrative examples. (Matrix groups appear as examples, which I thought was a nice touch.)

There are also lots of exercises; quite a few of them struck me as being in the easy-to-moderate range of difficulty, but there are also an adequate number that will challenge better students. The exercises are embedded in the body of the text, thus giving the student some guidance in precisely what ideas are useful for their solution. Many call for proofs, but there are also many useful problems that require the students to get their hands dirty with computations in specific examples. No solutions are provided, which I view as another pedagogical plus.

The book strikes me as being quite flexible. Instructors who prefer to focus just on the point-set theory could fashion a very nice one-semester course out of some or all of the five chapters in which this theory is discussed, but it is also possible to dip into some of the combinatorial theory at the expense, say, of some of the more sophisticated material in chapter 9. (The author states that he does not cover the material in this last chapter in his lectures.) My only regret with respect to topic coverage, and this is a relatively mild one, is the omission of any discussion of the fundamental group. Several other books (see, for example, the books by Croom or Gemignani) at least give some introduction to this issue; it might have made another nice section or two for chapter 9, and would have made the book that much more flexible.

A more serious problem, however, is the “one annoying feature” of the book that I noted in the first paragraph — a problem that is, unfortunately, not unique to this book, as Fernando Gouvêa’s review of Group Representation for Quantum Theory makes clear. This problem is, to quote that review, the lack of “one careful reading by a competent copyeditor”. Mistakes in English, many but not all involving the difference between singular and plural, are much too prevalent in Yan’s text. The table of contents alone, for example, with chapter headings like “Set and Map”, “Graph and Network”, “Surface” and “Complex”, is painful to read. Section titles include “Equivalence Relation and Quotient”, “Classification of Surface” and “Recognition of Surface”. One homework exercise asks the reader to prove that “the subset of a countable set is countable” and another requires proof that “the subspace of a regular space is regular”. On several occasions in the text, the author uses the word “alphabet” to mean “letters of the alphabet” (as, for example, in the question “Which alphabets are manifolds?”). The word “basis” is occasionally used to mean “bases”.

Some examples of poor English are serious enough to potentially cause confusion in beginning students. When the author states, for example, that “we see real invertible matrices do not form a connected space”, it is clear to a knowledgeable reader that the author means the set of all such matrices is not connected, but a beginner might be confused into thinking that no collection of such matrices can form a connected space.

Gouvêa noted in his review that the fault here belongs to the publisher, not the author, and, for the same reasons that he gave, I agree: Yan is not a native English speaker, and teaches in Asia; he should reasonably be allowed to rely on the publisher for competent editing. But, of course, it was presumably not the author who wrote “introduction of topology” on the back cover.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.