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Undergraduate Topology

Aisling McCluskey and Brian McMaster
Oxford University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Keith Jones
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In Undergraduate Topology: A Working Textbook, the authors attempt the difficult task of providing a compromise between the traditional approach to topology, consisting of lectures based on a monograph, and a student-centered approach more in line with inquiry based learning (IBL) or the “Moore method”. The guiding principle of IBL is that when students construct knowledge through the inquiry process, the learning is ultimately deeper. It emphasizes student solution and discussion of all facts beyond the definitions to develop the subject, rather than lecture or monograph delivery of core facts leading into student solutions to exercises. Point-set topology could be considered a poster-child for the use of IBL in college mathematics, due to its somewhat self-contained and abstract nature. Some may consider even the existence of a textbook to be counter to the principles of IBL, while many remain skeptical that it will work for their classes or that it isn’t a pedagogical fad. Extreme members of both camps will likely be put off by the premise of this text. However, instructors ranging between both ends of that spectrum who are interested in a student-centered approach supported by a clear and concise text may indeed find this book quite appealing. There is no doubt that this textbook is more student-centered than the standard monograph. Where a monograph is principally written with the subject in mind, it is clear that the authors of this book are thinking as much about their students and how they engage the material as the content.

The goal of offering a text that encourages students to construct their knowledge actively by emulating research mathematicians is admirable, and I think it is reasonably well accomplished in this book. The key distinction between this textbook and a monograph is that each chapter is broken into two parts: exposition and expansion. The exposition consists of what one would expect to see in a set of lecture notes, except (with few exceptions) no proofs or arguments appear. The material is presented very much in lecture note style as a long and well-organized list of definitions, lemmas, exercises, examples, notes, and last but certainly not least, warnings. It is a nice touch that the warnings appear on equal footing with the rest of the items. There do appear here and there brief paragraphs of prose nestled in this content, often written conversationally and with the student in mind. The proofs are missing from the exposition because, as the authors state in their message to the student, the students are expected, to the greatest extent possible, to construct these proofs themselves through study. Were the authors to have stopped here, the textbook might comprise exactly a set of notes akin to those many teachers currently provide students as a framework for an IBL-style topology class. The additional notes, warnings, and occasional prose further flesh out the exposition.

The expansion section, however, represents the compromise. The authors recognize that many (indeed most) students will not have the time or effort to independently realize all the proofs contained within even this short course, and so they present the majority of the missing arguments from the exposition within the expansion. Here as well, the authors recognize that this book is very much intended to be utilized by students, as the expansion content is formatted and communicated in a way they would expect a bright and diligent student might achieve “on a third draft”. These proofs are clean and terse, and indeed illustrative of what I think many of us wish our students would submit. One benefit is that this provides students with models for their work in a way that a monograph does not. (One might argue lecture notes should do the same; and if they do not, will these?) The expansion is neatly indicated graphically by having a “graph paper” style background, which does make it easier to navigate through the book. At this point the question might be asked whether the expansion defeats the purpose of IBL. My inclination is that there are many instructors who wish to take steps towards student-centered learning and desire that their students develop a mathematician’s habits of mind, yet are not prepared to “dive into the deep end” of IBL, and that those instructors will find value in the approach this book takes.

The textbook overall is slim, comprising only 140 pages, including a two-page index and 35 pages containing 79 exercises along with solutions to about half of them. It is made clear that students hoping to learn from this book should have a strong grasp of set theory and logic, and some real practice working with mathematical proofs. Additionally the authors expect students to have a “decent understanding of real analysis” and a first course in metric spaces. Certainly students should have some comfort with limits, sequences, continuity, and higher dimensional Euclidean spaces prior to using this book. The introduction does provide key results on set theory and functions that enable them to justly state that the book is generally self-contained, as well as presenting countability, Zorn’s lemma, well-ordering, the first uncountable ordinal, and the definition and examples of metric spaces. Chapter 2 dives into topological spaces, presenting open sets, neighborhoods, closed sets, closure, and subspaces. Chapter 3 discusses continuity and convergence properties, including sequences and first countability. It also includes a brief discussion of nets and filters, which does not appear necessary for the subsequent chapters. Here I did have one complaint: the authors provide a proposition involving compactness (a topological space \((X,\tau)\) is compact if and only if every filter on \(X\) can be refined to a convergent filter) prior to the definition and discussion of compactness. The authors do mention that this discussion is an extra topic, but a forewarning that compactness will be discussed in the following chapter might avoid some confusion. Chapter 4 is devoted to invariants including compactness, sequential compactness, local compactness, connectedness, and separability. Chapter 5 introduces Base and Product spaces, including Tychonoff’s theorem in both the finite and infinite case. The book concludes with a discussion of the separation axioms and Urysohn’s lemma, nicely bookending the course with an example of the second uncountable ordinal and the Tychonoff plank.

Its slim profile makes the book an affordable and attractive option for a one semester student-centered introductory course on point-set topology, or for a student working on a guided independent study. An instructor wishing to construct a fully IBL topology course could easily extract the exposition sections and exercises into a framework for a course, and provide a smaller selection of completed proofs. Conversely, this book may be appealing to professors wishing to teach a standard lecture yet desiring a book which both speaks to the students and provides a model for proof-writing. I believe that, used as intended, the textbook offers more than enough content to fill a semester, especially with the additional exercises; but it is not a thorough reference on topology. Notable topics that are missing from the discussion are identification spaces, path connectedness, and fundamental group. Additionally those looking for a combinatorial perspective, with discussion of Euler characteristic or cell complexes, will not find what they are looking for here.

Keith Jones is an Assistant Professor of Mathamatics at the State University of New York at Oneonta.

1. Introduction
2. Topological Spaces
3. Continuity and Convergence
4. Invariants
5. Base and Product
6. Separation Axioms
Essential Exercises
Solutions to Selected Exercises
Further Reading