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Bright students and topology
- March 2nd, 2007
Today, 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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Michi is a PhD student working in homological algebra. This blog is his outlet for texts with some manner of thought put into them. Over at his LiveJournal intimate details and streams of consciousness might be found.
Not all here is mathematics. All here, though, are my personal thoughts and opinions. Please read the about page (linked above) for more details.
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9 People had this to say...
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.
These are frighteningly good students. They ended up with me because they have already outrun every single teacher at their - mathematically profiled - school. One of them is reading university-level analysis just for the fun of it.
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’m going to push for fundamental groupoids
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.
Good luck! They really are specialised school students, I’d never even heard of topology until the final year of my degree…
At the University of Chicago we have this program called YSP which brings in gifted (and I mean it) students to learn fancy math. When I was there we were teaching algebraic topology (through Knot Theory) and the theory of computation to 11th and 12th graders. And I had only studied algebraic topology the year before at Chicago.
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!
Jesse: Yeah, that sounds similar to what I’m doing. My students are younger, fewer, but basically equally smart.
Armstrong’s “Basic Topology” sucks! Please use another book! The explanations in that book are horrible. Armstrong was the first book that I used when I tried to learn topology, and I was really turned off from the subject until I found Hatcher’s book.
Kevin: Don’t worry. So far, we’ve been using Jänich almost exclusively, and once we get to the algebraic topology side, I’m going to want to look for Ronnie Brown anyway, since I want to introduce it along the lines of fundamental groupoids.
[…] with my bright topology 9th-graders, we discussed homotopy equivalence of spaces and the fundamental groupoid. In order to get the […]
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