Bright students and topologyToday, I started an experiment together with the local specialised secondary school. I’ll be taking care of two of their brightest students, meeting them roughly once a week, and taking them on a charge through algebraic topology. At the far end shimmers knot theory and other funky applications; and on the way there, I hope for many interesting spinoffs and avenues.
They got, today, Armstrong’s Basic Topology, and an extract from the German topology book by Jänich, and on monday, we shall go over the formal definition of topological spaces, and of continuous functions together.
I plan to keep updates on our progress here on the blog – with the questions I send them off with each meeting as well as some sort of discussion about how this setup is working out, if at all.
For the first trip, the questions I dumped in their laps were:
- What topologies are possible on the set {0,1}?
- What topologies are possible on the set {0,1,2}?
- Which are the continuous functions between the topologies above?
- Give an example of a continuous and a discontinuous function each for the following cases
with the standard topology on 
with the standard topology on 
with the discrete topology
- with the finite-complement topology
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These must be really bright students. Starting with general topology seems like a tough place to start!
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.