Several years ago I had the pleasure of teaching an introduction to proof course with Michael Starbird’s *Number Theory Through Inquiry* (co-authored by David Marshall and Edward Odell). Now I often teach an introduction to proof course centered on point-set topology. It seemed natural to open this text to see what it has to offer.

*Topology Through Inquiry* includes a substantial amount of subject matter. There is an entire course in point-set topology, along with the second half of the text devoted to algebraic and geometric topology. This second half includes chapters on the classification of surfaces, the fundamental group, covering spaces, and several pertaining to homology. The text would be suitable for my current one-term course, concentrating just on the first half, or for a year-long course. For a student wishing to pursue an independent study course, this would be a wonderful text either as an introduction to topology or for further study after an introductory course. In fact, in the preface, the authors provide several ways to navigate their work in various contexts like these, including some sense of which sections are fundamental to study.

Two things about the presentation stand out to me. First, I was surprised to see that the topic of continuity does not make an appearance until Chapter 7. This is after discussions of separation properties and compactness, and a chapter entitled “Countable Features of Spaces” which introduces first and second countable spaces. For the category theorist in me, once the basic objects of study have been defined, I want to know what sorts of action I can take with those objects, ie, what the relevant maps are, and the wait here felt very long. Second, almost the last quarter of the text is about homology. The approach is to work with Z2 coefficients first, and use this to prove the famous theorems (the Borsuk-Ulam Theorem, the Ham Sandwich Theorem, etc). Then Z coefficients are introduced. Mainly simplicial homology is used, although cellular homology and singular homology do make brief appearances. These choices surprised me, in a pleasant way. Simplicial homology is the underpinning of the computational algorithms used in the burgeoning field of applied topology, and after working through these chapters, the reader will be well-positioned to embark on a study of that field.

As expected, exercises comprise most of the text. Actually, that’s not quite true. There are many theorems, the proofs of which are left to the reader. There are also many exercises that ask readers to develop examples, think about the necessity of certain assumptions, think about generalizations, and so forth. The text is more than just a list of things to prove though, rest assured. The connective tissue of the book establishes the context and a framework for the results. The authors are particularly adept at including Effective Thinking Principles throughout the text. These strategies are set off in the typeset so that the eye is drawn quickly to them. They embody what most of us exhort our students to do: create examples, notice assumptions, generalize theorems, etc. Many are strategically placed so as to clearly motivate particular exercises; this direct connection is often lacking in a traditional set of textbook problems.

As I write this, we have just finished a term of online learning. The experience was a stark reminder that, for many of our students, learning how to learn is a large part of learning. What we do naturally in our synchronous interactions with students by asking them to remind us of a definition, by working out basic examples, by connecting one theorem to the next…these are exactly the pieces that our asynchronous classrooms easily miss. It is for these reasons that a book like Topology Through Inquiry can make an impact. Effective Thinking Principles need to be made explicit to our students, and repeatedly so. The practice of actively reading a text is one we need to help our students cultivate. Here is a great resource. And of course,

*Topology Through Inquiry* goes further: it provides the framework but requires the reader to construct the body of knowledge for herself.

Michele Intermont is an Associate Professor of Mathematics at Kalamazoo College.