Starting from tomorrow, I will be tutoring some undergraduate students following a course in general topology. I am looking for examples motivating the importance of topology in mathematics which can be explained without too much difficulty using concepts of other areas of mathematics (or physics) they have already treated (those areas would be mainly analysis, complex analysis, linear algebra, a little graph theory, some numerical methods for maths, and classical mechanics, electromagnetism, special relativity, some QM and a little statistical physics for physics). I have tried looking around, but I have found little that would motivate me to follow such a course. Do anybody have some nice example?
Note: I will of course explain to them that without topology they'll be able to do very little advanced mathematics (e.g. functional analysis, differential geometry, …)
EDIT: Ok, I gave as examples Tychonoff's theorem, Brower's fixed point theorem and the Jordan curve theorem.
I would like to keep this question alive, for personal interest. What are interesting (not too hard) applications of topology in other areas of mathematics?
Best Answer
Well, an idea would be to talk about results of Real Analysis and how topology generalises them, or how topological methods make their proofs much easier. You could consider the Intermediate Value Theorem for instance, or how the Lebesgue Number Lemma makes it very easy to prove that if $f: X \to Y$ is a continuous function between metric spaces $X,Y$ where $X$ is compact, then in fact $f$ is uniformly continuous with respect to the topology induced by the metric. Tychonoff's theorem might also be a nice one, or the converse of the Closed Graph Theorem. My guess is that you should connect it to metric spaces mostly, for they will be mostly familiar with that. If, of course, they are more advanced, you can consider talking about how topology shows that it is impossible to find a homeomorphism $f: [0,1] \to S^1$ or other similar constructions. I can keep going but I am afraid that it would be pointless if any of my previous suggestions is not readily utilisable. Best of luck!