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Elements of Topology

Tej Bahadur Singh
Chapman & Hall/CRC
Publication Date: 
Number of Pages: 
[Reviewed by
Mark Hunacek
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Back when I was a junior in college, I took an introductory point-set topology course based on the first half of the textbook Introduction to Topology and Modern Analysis by George Simmons. (The second half of the book is an introduction to functional analysis at an undergraduate level.) I thought at the time, with the certainty that only a 20-year old kid with two whole years of college under his belt could muster, that this book must surely be an example of a perfect undergraduate text.

Now, forty-some years and a lot of other mathematics books later, I realize that although Simmons’ text holds up remarkably well as an example of lucid exposition, it is hardly perfect. Its biggest problem, I think, is that it suffers from several serious sins of omission: there is no mention in the text of the Möbius strip or the Klein bottle, for example, or, for that matter, any discussion of quotient spaces at all. In addition, although the Baire Category Theorem is stated and proved, there is no real indication of any of its applications. (I agree with Chekhov’s admonition that one should not bring a loaded rifle on stage if it isn’t going to be fired.)

The book now under review, which arises from the author’s experiences in teaching both undergraduate and graduate courses in topology, does not commit the errors noted above. I can’t think of any significant topic that I would like to see covered in an introductory topology course that is not discussed here. Not only does this book cover the standard examples of quotient spaces and mention applications of Baire category (both in the text and in the exercises, where, for example, the author requests the reader to prove the existence of a continuous, nowhere differentiable function), but it also contains very nice discussions of other topics that I think really enhance such an introductory course. I particularly liked the extensive discussion of transformation groups, for example, but then again I’ve always had a weakness for matrix groups, and I can’t help but feel that a student would really enjoy seeing that the group SO(n) of n × n real matrices of determinant 1 is not only a group but also a path-connected topological space.

Perhaps it is unfair to compare the coverage of this text with Simmons’, because this text covers more than enough material for a full year course in topology (including an introductory look at algebraic topology), whereas the book by Simmons was intended for a one-semester course followed by an introduction to functional analysis. Nevertheless, there is no denying that all of the extra material included in this book allows for a degree of flexibility that is not present in Simmons.

The topics covered can be easily summarized. The book begins with a chapter on topological spaces (the standard introductory material on set theory is relegated to an appendix) which, following what I think is the most logical approach, begins with the definition of a metric space, and talks about those to the extent that the definition of a topological space is then seen as a natural generalization. The first chapter introduces some basic terminology connected with topological spaces, and the second chapter discusses continuity and the product topology — finite products first, then infinite ones.

The important concepts of connectedness and compactness are discussed in chapters 3 and 6, respectively, with the two chapters in between addressing convergence (after illustrating the inadequacy of sequences in general topological spaces, the author discusses filters and nets) and first and second countable topological spaces (which ties in with the previous chapter in that sequential convergence is a much more useful concept in these spaces). The chapters on connectedness and compactness cover all the usual ideas: a characterization of connected sets on the real line (i.e., intervals), the difference between connected and path-connected spaces (with everybody’s favorite example — the closed topologist’s sine curve — of a space that is the former but not the latter), the Heine-Borel theorem, Tychonoff’s theorem (proved via nets), compactness in metric spaces, the one-point compactification, and more.

The seventh chapter addresses quotient spaces (mentioning the standard examples that Simmons’ book does not) as well as other forms of constructing new spaces from old ones: cones, suspensions, joins, wedge products, and smash products, all illustrated with nice pictures and clear descriptions and examples. These first seven chapters are described as the “core” of the book, suitable for a one-semester upper-level undergraduate course; I suspect there is a bit more here than could be covered in such a course, and if I were teaching such a course I would probably omit some topics and try and sneak in some of the later material on matrix groups. The book is written so as to allow such flexibility.

The next four chapters address additional topics in point-set topology: separation axioms (including regular and normal spaces, the Urysohn Embedding Theorem and the Tietze Extension Theorem, and the Stone-Cech compactification), paracompactness and metrisabiity (including the Nagata-Smirnov metrization theorem), completeness (this is the chapter the Baire category theorem appears in), and function spaces (including the compact-open topology, equicontinuity and the Arzela-Ascoli theorem, proved in a fairly general form).

This is followed, as I mentioned earlier, by two chapters on topological groups and transformation groups, written at a fairly high level but containing some nice concrete examples, such as the classical matrix groups. Even the quaternions make an appearance here, which doesn’t often happen in elementary topology texts.

The remaining two chapters of the text introduce the reader to the beginnings of algebraic topology, namely the fundamental group and covering spaces. Homology is not mentioned, which seems entirely appropriate; I have doubts as to whether homology theory can really be successfully taught to undergraduates, and believe instead that it belongs in a full-fledged graduate course on algebraic topology, with a book devoted entirely to that subject, such as Hatcher’s Algebraic Topology.

These two chapters instead focus on the (first) fundamental group (higher homotopy groups are also not discussed) and the basic applications and calculations: the Fundamental Theorem of Algebra is proved, for example, as an application, and the fundamental groups of a number of spaces are established, as is the Seifert-Van Kampen theorem. Covering spaces are discussed to the point where it is shown that for a “reasonably nice” space X, there is a covering space corresponding to every subgroup of the fundamental group of X. (Obviously the author makes things much more precise than this quick summary does.)

The selection and organization of material in this text is fairly similar to that in the well-known Topology by Munkres, but there are also some substantial differences between that book and this one. Munkres spends more time on algebraic topology (he covers the classification of surfaces, for example) but does much less with topological groups (his coverage is limited to some supplementary exercises at the end of chapter 2, and he doesn’t have nearly the kind of discussion of the classical matrix groups that Singh provides).

The overall tone and level of sophistication in this book is roughly comparable to that of Munkres, and is considerably higher than in Simmons’ text. The overall writing style is, however, sufficiently clear that there is no question of the suitability of (at least much of) this text for undergraduates. The pedagogical value of the book is also enhanced by the presence of quite a number of exercises of varying levels of difficulty (solutions to which are not provided), and also a substantial number of detailed examples in the text itself.

One mild complaint: it did seem to me that the Index could stand some improvement. (The Index in Munkres is 19 pages long; the one in this book is 4.) My initial attempt to locate the Baire Category Theorem, for example, was frustrating because neither “Baire” nor “Category” appear where I expected them (under “B” and “C”); after some additional perusal I did find a reference under “S” (“spaces, Baire”). This trick doesn’t always work, though: Lindelof spaces appear under neither “S” nor “L”. Likewise, I couldn’t find any reference in the Index to the topologist’s sine curve, and also encountered similar frustration on one or two other occasions.

Let me, however, not end a very positive review on a negative note. This text presents a considerable amount of material in a clear and accessible way, and should be carefully considered for textbook adoption by anybody teaching a course in point-set topology.

Mark Hunacek ( teaches mathematics at Iowa State University. 

Topological Spaces
Metric Spaces
Derived Concepts

Continuity and Products
Product Topology

Connected Spaces
Path-Connected Spaces 
Local Connectivity

Hausdorff Spaces

Countability Axioms
1st and 2nd Countable Spaces
Separable and Lindelöf Spaces

Compact Spaces
Countably Compact Spaces
Compact Metric Spaces
Locally Compact Spaces
Proper Maps

Topological Constructions
Quotient Spaces
Identification Maps
Cones, Suspensions and Joins
Wedge Sums and Smash Products
Adjunction Spaces
Coherent Topologies

Separation Axioms
Regular Spaces
Normal Spaces
Completely Regular Spaces
Stone-Čech Compactification

Paracompactness and Metrizability
Paracompact Spaces
A Metrization Theorem

Complete Spaces
Baire Spaces

Function Spaces
Topology of Pointwise Convergence
Compact-Open Topology
Topology of Compact Convergence

Topological Groups
Examples and Basic Properties
Direct Products

Transformation Groups
Group Actions
Orbit Spaces

The Fundamental Group
Homotopic Maps
The Fundamental Group
Fundamental Groups of Spheres
The Seifert-van Kampen Theorem

Covering Spaces
Covering Maps
The Lifting Problem
The Universal Covering Spaces
Deck Transformations 
The Existence of Covering Spaces

Appendix A: Set Theory
Appendix B: Fields R, C and H