Teaching Topology: A Preview of Shaun Ault's "Understanding Topology"

As an educator and research mathematician, I feel that my primary job is to share my knowledge and expertise, to shed light on difficult concepts, and to encourage students to continue discovering beyond what’s presented in the classroom.  When I began work on the textbook, Understanding Topology: A Practical Introduction, it was with these aims in mind.  Above all, I wanted this textbook to sample a wide range of topics that fall under the broad category of topology while remaining rigorous enough for students to understand how mathematicians to do real mathematics.  I wanted students to encounter new and wonderfully strange topological spaces and have the tools in hand to analyze them.  Furthermore, I wanted students to see how this very abstract field of mathematics could be useful to other areas of math and science.

Understanding Topology began life as stack of messy lecture notes that I had prepared for an undergraduate course in Topology at Valdosta State University.  The lectures were based on material cobbled together from a variety of sources, including The Knot Book by Colin Adams, Beginning Topology by Sue Goodman, A Combinatorial Introduction to Topology by Michael Henle, The Shape of Space by Jeffrey Weeks, and of course the classic Topology by James Munkres.  Eventually, after having revised, edited, rewritten, and added to the lecture notes over the years, I finally decided to compile and typeset them, mostly so that I would have a clean set of notes to work from in future Topology classes.  However, it didn’t take long to realize that what I was actually doing was writing a textbook.

There are many other great undergraduate textbooks on Topology on the market right now, but what sets my book apart from the others is a combination of features.  First and most importantly is the breadth of the topics – from Euclidean and metric topology, to vector spaces, to abstract topological spaces and surfaces, to maps and knots, to elementary homotopy and homology theory.  Each section contains numerous examples and clear exposition alongside formal definitions and rigorously proven theorems.  With over 180 figures together with many other diagrams, tables, and illustrations, to aid students’ visualization, and almost 300 exercises, many of which have full solutions in the back of the book, to allow students to practice the concepts, this text truly prepares students to understand topology.

The book is also versatile.  There is an appendix on set theory and functions, which may be presented in lecture or assigned for independent study.  A typical semester of study may focus only on Parts I and II (Euclidean Topology and Abstract Topology with Applications).  For a more fast paced course, Part III (Basic Algebraic Topology) could be included, or this part may be used as a component of a separate advanced course (for example, as a seminar after students have taken Abstract Algebra).  The basics of group theory and linear algebra are included as an appendix to the textbook as well.

Of course no textbook can please everybody or work in any given situation.  Hard choices had to be made about what to include and what to leave out.  In its current form, the book does not address vector fields on general surfaces (although a later edition of the book might fill that gap).  There is also the issue of the incredibly important but surprisingly difficult to prove Jordan Curve Theorem.  While this textbook uses the Jordan Curve Theorem as a basis for a number of other results, I could not include a proof of it without taking the reader too far from the core focus of the book.  Many other textbooks follow this same route, which I feel is justifiable in an undergraduate setting.

I hope that Understanding Topology lives up to its title.  Check it out, especially if you are considering teaching a course in topology at the undergraduate level.

Shaun V. Ault is an associate professor at Valdosta State University. He is also the author of Understanding Topology: A Practical Introduction