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.

You have describe my experience exactly! I remember that I had just taken a Real Analysis class and left thinking the class was really dumb. What are these silly contrived metric spaces, what’s the point of compactness or completeness, is this weird epsilon-delta continuity thing really necessary? The following semester, I took Topology and it all clicked. I have never needed to use Hausdorf or Hilbert or connectedness. But when I took semesters of complex analysis, linear algebra, abstract algebra, stochastic processes, and functional analysis, I had a much easier time than my classmates — and I believe it was because I could see the relationships and the bigger picture. Topology was my mathematical awakening. Thank you for this post!

Is there a text you’d recommend for self-study?

Jake, I don’t know of a text I’d recommend.

I’m not recommending topology per se. My topology class had a big impact on me because of the circumstances. It could have just as easily been a course in a different topic.

It’s hard to have an epiphany from self-study. A lot of the breaktrhoughs in my thinking have come through interaction with other people. If you can’t take a college course, maybe hire a tutor, preferably one you meet with face to face.