Next semester: I have a usual load of two calculus courses: calculus III (polar coordinates up to the three main integral theorems…or merely Stokes theorem), “applied calculus II” (the follow on to the course sometimes called “business calculus” or “brief calculus”) and an undergraduate course in topology.
The latter will be fun, but a major time suck, and it will be challenging to teach.
Here is where the challenge comes: though I got my Ph. D. in 1991, I have NEVER taught our topology course. But I have published many papers in this area; I really know the stuff well. But I’ve never tried to explain it to non-research mathematicians.
So, how does one teach something that someone has “done” for 26 years but has never tried to explain to a beginner?
Then there is the challenge of the course itself.
On one hand, there is some basic stuff that one needs to know to advance in mathematics (e. g. the point set and the metric stuff).
On the other hand: the more interesting stuff (the tori, Klein bottles, Mobius bands) is really a collection of “parlor tricks” unless it is presented mathematically, and one needs the basic stuff to do a proper job.
Add to that the non-intuitive notation and jargon:
So: I’ll probably start with some set theory, some metric space theory (say, at the epsilon-delta calculus level at first), some topology of and then go to the more abstract stuff, with a lot of examples.