Topology: An Invitation

Topology traces its rich history all the way back to centuries-old ideas of Leibniz and Euler. The precise definition of a topological space is of course of a more recent vintage, and the evolution of the subject across the intervening 300+ years is not only a compelling story, but also a useful source of motivation and understanding from a purely mathematical point of view. This is perhaps indicative of why primary historical sources can be used to great effect in undergraduate mathematics courses more generally, as advocated by the algebraic topologist David Pengelley and others. The book under review enters the relatively crowded marketplace of introductory topology texts and manages to make its own mark with its uniquely historical perspective. Each topic is, in the words of the author, “spiced up with a dash of historical development.” This is a bit of an understatement. There are numerous quotes taken from original source materials (often featured in the original language, and then translated) and relevant biographical snippets from the lives of various mathematical luminaries (Cantor, Lebesgue, Poincaré, etc.) throughout the book. The result is a fresh and thoroughly charming contribution to the topology literature.

 

A look at the table of contents reveals a largely standard list of topics. Preliminaries on metric spaces lead into general topological spaces, continuity, compactness and connectedness, and the separation axioms, with metric spaces being revisited later in a chapter on completeness. Toward the end the author introduces combinatorial topology, homotopy, and the fundamental group, providing the reader with an on-ramp to

algebraic topology in the style of other authors such as Gamelin & Greene or Munkres.

 

The brief real-analysis style review of the Intermediate Value Theorem (IVT) and its consequences in Chapter 1 is a friendly way to begin an exposition of a technical subject like topology. The author characterizes this introductory chapter as an “apéritif” for the “gourmet course to follow.” How inviting! The phrase “An Invitation” is literally in the book’s title, so this should be no surprise. The first sentence of the chapter is the statement of IVT in Bolzano’s own (translated) words. The power of IVT is then put on display with corollaries such as the ham sandwich theorem (called the “dosa theorem” by the author, in an endearing nod to the food culture of his home country) so that the reader is motivated to uncover the presumably even more powerful topological concept underlying IVT later in the text. After a nice batch of exercises, the chapter then circles back to Bolzano with some biographical notes on his life and his influence on mathematics. This is a great start, and thankfully, many of the attractive features of Chapter 1 are perpetuated in subsequent chapters.

 

Even though the author confesses that he “is no historian of mathematics” in the preface, it is his seamless integration of historical references within each and every chapter that drives the narrative and makes the book unique. Consider Chapter 5 on compactness as an example. Like Chapter 1, this chapter opens with direct quotes from past mathematicians, this time featuring Fréchet remarking on the importance of compact

sets in 1906, and Vietoris giving the first definition of compactness for general topological spaces in 1921. The references to primary historical sources continue throughout the remainder of the chapter and they always support the logical flow of the material in interesting ways, as do the biographical notes on Borel, Heine, Lebesgue, Cantor, and Vietoris that appear at the end. One learns, for example, that Vietoris submitted his thesis shortly after being a prisoner of war in World War I, and lived to be nearly 111 years old!

 

Any faults to be found with this book are mostly typographical. While the book features an extensive and impressive bibliography, the bibliography itself is split apart by chapter, resulting (for example) in J.L. Kelley’s classic topology text being referenced with five different reference numbers across eight chapters. A single bibliography appearing at the end seems less confusing and more appropriate for this particular book.  While the book includes plenty of figures, the quality of the figure rendering is inconsistent. The mathematical text overlaying the figures is sometimes properly typeset in LaTeX, sometimes not, and certain figures could be enlarged to avoid mathematical symbols being jammed into tight spaces which would improve readability. While the author’s use of LaTeX symbols is actually quite creative in some spots (see, for example, the symbol used to denote the gluing of two continuous maps in Chapter 4), problems occasionally arise when certain Roman letters are used mathematically.  Take as an example the book’s convention of using the letter p to denote a covering map throughout Chapter 15. The author’s choice of typesetting for this p symbol results in p being italicized when appearing in the statement of a theorem while remaining un-italicized elsewhere, making it hard for the reader to consistently track the symbol’s meaning. Other more traditional function symbols like f and g are typeset in the usual way throughout the book and thankfully do not exhibit this same inconsistent appearance. 

 

One final comment to be made concerns the exercises. While each exerciseset contains a large number of well-constructed problems, many of the exercises lack the usual commands to the reader such as “Show that” or “Prove that,” and are instead given simply as statements that the

reader should know to prove. Perhaps this was done to be more succinct? Whatever the intent, including the usual commands to the reader would greatly improve the readability of the exercise sets without significantly lengthening the book.

 

The discussion of topology with an historical bent is not entirely novel (see, for instance, the tome History of Topology edited by I.M. James), but it is quite novel for an introductory topology text. This book can and should serve as a template for how introductory texts in other subjects might be “spiced up” with references to primary historical sources and biographical snippets that allow the reader to explore the material seemingly hand in hand with the mathematical giants who created it.

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Don Larson is an Assistant Professor of Mathematics at the Catholic University of America. His main research interest is in algebraic topology.