These kids are super-smart. For example, in the class was the great-grandson of Max Zorn and the son of a Fields Medalist. Makes me feel like a slacker for not learning this stuff until I was 21 or 22. Hah!

Cool! It does mean that they’ll have trouble referring to books but I agree it does make more sense. Since realising that the 15 puzzle is more naturally described by a groupoid than a group I’ve thought that groupoids ought to be introduced at an earlier level.

I’m going to push for fundamental groupoids instead of fundamental groups, since I think that if you don’t work from the point where the algebraic terminology is floating around anyway, it might just be a more natural way of looking at things.

And the general topology stuff, I use mainly now in the beginning, in order to gauge their speed, and to set the stage for the homotopy theory I want to bring in.

I’d have gone for an approach that assumes intuitions about continuity first, developed a bit of knot theory (say), and then gone back to general topology. Fun concepts like fundamental groups, Reidemeister moves and rational tangles can be introduced without a solid grounding in the notions of continuity, and these can then help to motivate general topology later on.

But if your students can keep up, then throw ‘em as much finite-complement topology as they can handle.