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Topology: An Introduction

Stefan Waldmann
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[Reviewed by
Keith Jones
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Stefan Waldmann’s Topology provides a fast-moving introduction to the core concepts of topology. The author indicates in the preface that the goal for this particular book is to provide a minimal introduction to topology necessary in order to move on to more advanced mathematical topics, such as functional analysis or differential geometry, or to physics. Indeed, the mathematics covers the standard topics, with proofs, in a trim 125 pages. Moreover, the author indicates that a personal preference for applications to functional analysis led him to culminate in a discussion of continuous functions, including proving Urysohn’s Lemma, Tietze’s Theorem, the Stone-Weierstrass Theorem (with a section on \(C^*\) algebras), Arzelá-Ascoli (non-generalized), as well as a chapter on Baire’s theorem. The author’s view of topology as part of a broader context is apparent in some other ways throughout the book. Within the discussion of continuous maps, for example, the author brings up the notion of a category and includes a handful of examples, in order to point out that topological spaces form a category with homeomorphisms as the morphisms. Group actions are recalled and described in order to provide examples for discussing the quotient topology. Section 3.3 of the text is devoted to a primer on topological manifolds. All of these help give the reader to better picture how topology might fit into the landscape of mathematics. Advanced topics in other directions, such as algebraic topology, are not covered.

The textbook does (unsurprisingly) demand a strong background in set theory and logic. The first section does contain an overview of metric spaces in order to motivate the more general definitions to come, and there is a very brief appendix on naive set theory. But this five page appendix, while beginning with most basic concepts, movies quickly to requisite properties of images and preimages of maps before presenting a one-page introduction to the axiom of choice and Zorn’s lemma. Thus, it is far from a proper introduction to even naive set theory, and so the author’s honesty is appreciated when he aptly names this appendix “Not an Introduction to Set Theory.” Students will certainly need to have already had a more proper introduction to concepts of set theory to be comfortable with this book. For example, equivalence relations and equivalence classes are applied in the discussion of quotient spaces with no formal introduction provided in the book, and there are no exercises related to the basics of set theory. This is not a book that leads a student gently into abstraction, but rather it assumes a high level of mathematical maturity and comfort with proof.

The first chapter is a very brief introduction, including a few words of motivation for the subject along with an overview of the structure of the book. Chapter 2, introducing the basic notions, up to and including connectedness and the separation axioms, is by far the longest chapter at about 35 pages. It might in-and-of-itself make a good framework for an undergraduate independent study or slower paced undergraduate course. Chapter 3 introduces standard constructions such as the product and quotient topologies and their generalizations, the initial and final topologies. Chapter 4 discusses convergence in the context of nets and filters. Sequences are only mentioned as a special case of nets and are not discussed directly. The use of filters pays off in chapter 5, which is devoted to compactness and brings together previous facts and exercises to quickly prove Tikhonov’s theorem.

As mentioned above, chapters 6 and 7 are devoted to proving some of the major theorems of general topology, which find application in a variety of contexts. The discussion of the Stone-Weierstrass Theorem in section 6.2 includes an introduction to \(C^*\)-algebras and is is couched in this language. There are roughly 79 exercises in total, appearing at the ends of each chapter, with about one-third of them appearing in chapter 2, and with 8–12 exercises for each of chapters 3–7. As mentioned, a number of these exercises find application in proofs later in the book.

Overall, this is a well put-together introduction to most of the fundamental topics of topology necessary for the study of advanced mathematics. A student well-versed in the notions of set theory and metric spaces should find this an appropriate introductory resource for topology. Additionally, the book provides a very nice and conveniently compact reference for the standard topics of general topology, and this compactness does not come at the cost of omitting proofs. It might be particularly relevant to students moving into functional analysis, given the author’s choice of topics and approaches. Introductions to topics and remarks are kept brief, but are sufficient to motivate and flesh out the mathematics. While the book is not exactly teeming with examples, there are certainly enough, and with a wide enough variety, to enable the diligent student to obtain a broad view of general topology and many of its applications.

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