Most useful math class

A few years ago someone asked me what was my most useful undergraduate math class. My first thought was topology.

I have never directly applied topology for a client. Nobody has ever approached me wanting to know, for example, whether two objects were in the same homotopy class. But I believe topology was one of the most important classes I took for three reasons.

First, I learned how to prove things in that course. It was a small, interactive class with an excellent teacher (Jim Vick). I might have learned the same techniques in a different class, but for me I learned them in topology.

Second, the course built my confidence. I was apprehensive about taking the course because I knew nothing about it. The little I'd heard about topology-stretching coffee cups into donuts etc.-made me wonder what a class could possibly be like. I proved to myself that I could jump into something unfamiliar and do well.

Finally, the course gave me a solid foundation for analysis, and analysis I have applied more directly. I got a thorough understanding of foundational ideas like continuity and compactness, and a foretaste of measure theory. The course also provided my first brief exposure to category theory. To this day, my Pavlovian response to a mention of functors is to think of the fundamental group of a topological space.

I look back on topology the way many look back on a classical education, something not directly useful but indirectly very useful.