At first glance, this book looks like a standard point-set topology book. The chapters and sections are laid out in a nice way to help guide the reader who has little to no knowledge of topology through a year’s worth of material. The author starts with sets and functions, follows with a typical layout of point-set topology, and ends with a brief introduction to the fundamental group. Despite its standard set-up, however, this book is far from the normal topology textbook because it forces students to delve deep into understanding topology and also mathematical thought.
At a second glance, this book looks like an outline of a standard point-set topology book. Very few theorems are followed by proofs. Examples are scarce and are used to motivate the material instead of illustrating concepts. As the author states in the preface, however, this book is more than an outline. It is a discussion. It is important to know how the author intends the book to be used in order to truly utilize it. The author aims to have the reader not only learn the rigor of topology by writing proofs and doing exercises, but he aims to have the reader think about topology as mathematicians do — visually.
This book reads more like a story of topology than it does a textbook. The writing style is familiar and very easy to follow. Readers can rely on their intuition to understand complicated ideas with the help of simple examples to motivate and understand more complicated topological concepts. Exercises are embedded within the sections instead of listed at the end of a section. This feature is nice because it helps the reader understand the relationship of an exercise to the material which can difficult to see sometimes when an exercise is separated from the context.
This book would work well for a reading course, independent study, or even possibly a graduate level point-set topology course, because there just simply is not enough time in a typical yearlong or semester long undergraduate course to provide missing proofs for all of the theorems in class. The reader must get their hands dirty and devote a lot of time and effort into “completing” the text. It should be noted, though, that this is a key feature of the book, by design. The author wants the reader to try to prove theorems in order to learn how to write mathematics. Also, the author does have realistic expectations. Difficult proofs are given and when necessary, the author does provide hints but they can be vague like, “think about definitions”. Overall, this is a solid topology book for what it was written to be.
Whitney George is an Assistant Professor at University of Wisconsin- La Crosse whose training is in Contact Topology and Mathematics Education.