As of today, I’m taking over teaching general topology for an ill
colleague. I’ve taught a course like this a few times before, both at
Harvard and here at Edinburgh (but never at MIT, somehow). In 2020, I
even put some topology lectures online
(some of which are kind of hokey, but you can’t expect much from a guy
who’s been stuck it his house for weeks). ^{1}

Anyhow, general topology is a class I really enjoy, and I’ve been reflecting why.

One reason is that there are so many different ways to tell the story. You can define a topology as a system of opens, or a system of closeds, or as a closure operator. Each of those tells you something about what topologies capture. It’s just fun to explore the different narratives.

General topology is usually the first time students encounter real

*beasts*in mathematics. The definition of a topological space is so extremely general that you need to restrain the concept with a whole bunch of conditions to keep wacky “pathological” topologies out.But then a lot of those beasts turn out to be friends. The Cantor space seems weird and unnecessary, like a trick you use to annoy students. Then you learn about the -adic integers. Eventually the Cantor space just starts to feel like what you have to do sometimes to pass from finite discrete structures to infinite ones.

More generally, you start to appreciate the following idea: when you have an algebraic structure that isn’t finite or finitely generated, your only hope of dealing with it responsibly is to incorporate a topology. You see this with the -adics, but you also see it with absolute Galois groups, Lie groups, topological vector spaces, infinite-dimensional representations, etc., etc.

Topology resembles a viscous liquid poured over the rough surface of students’ understanding. Slowly the idea seeps into all sorts of nooks and crannies and gaps in students’ mathematical worldview. The rough surface over time becomes much smoother and much more navigable. It’s a powerful thing, but appreciation for it develops only gradually.

But as a lecturer I have to *start* somewhere, and first
lectures freak me out. The students and I are strangers (usually), and
yet I’m implicitly asking them for the most precious resource they have
– their attention. 50 minutes is a lot of their time, but it’s not
nearly enough of mine. I have to work out what they already know, what
they don’t yet know, what they think is important, what they think is a
waste of time, what they think is interesting, what they think is
boring, etc., etc. – and I *have to be prepared to try to change
their minds about some of this*. None of this includes actually
communicating the mathematics itself correctly!

It doesn’t help that students in the UK generally expect more
organization and more structure than I can deliver. I think lecturers
here are usually much more orderly and formal. Regimented, even. But
long ago, I gave up on the idea that I could ever give lectures that are
both thorough and comprehensible to most of my audience. Instead, I come
in with a few points I want to underscore, but my goal is a
*conversation* with the students. I gently steer that
conversation, but I don’t press an agenda too hard. Students have
questions, confusions, concerns – those are generally my first priority.
It’s a high-risk, high-reward approach to teaching: on a good day, my
lectures are dynamic and interactive; on a bad day, they’re chaotic and
annoying.

Naturally I’m incredibly nervous beforehand. Just for fun, I looked at my heart rate in the moments just before class started. (I have one of those watches that records that.) It was about the same as when I’ve been cycling up a gentle hill for five minutes.