Motivation for the importance of topology

Question

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?

Answer 1

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!

Answer 2

the point is that topology makes it easier to answer questions which are only a little less informative than many of ones natural questions, in many situations. E.g. instead of actually finding solutions to an equation, which is usually hard, one asks instead to show the existence of solutions, or to count the number of solutions in a weighted sense. Thus the intermediate value theorem provides an existence theorem in cases where the theorem of Abel tells us there is no formula in radicals for a root of a polynomial. Adding the mean value theorem we get estimates on the number of solutions. Similar two dimensional versions allow the fundamental theorem of algebra to be proved, again guaranteeing solutions to polynomial equations and counting their number. The general theory of degree of mapping generalizes this technique and even extends to infinite dimensional spaces. A nice source for that theory is Perspectives in Nonlinearity by Melvyn and Marion Berger. The theory of singularities of vector fields is another example of applying topology to questions of existence of solutions. I.e. the Euler characteristic of the sphere being 2 prohibits the existence of a never zero tangent vector field on the sphere. Cauchy's theorem also allows the concept of simple connectivity to be invoked to conclude existence of complex logarithms in certain regions. The list is long......

Answer 3

• In functional analysis, the notion of weak topology is a great example of application of "weird" (not metrizable) topology : arguments of weak compacity provide very quickly the existence in a lot of variational problems (including for example the Lagrangian formulation in classical mechanics or limited relativity).

• The "Hairy ball" theorem has a funny consequence : at every given time there are two antipodal points at the surface of the Earth that have exactly the same temperature and pression.

Answer 4

My professor's example: He used right index and thumb to form a ring, and left index and thumb to form another, interlocking the right hand ring. He asked if it was possible to deform him such that we unlock the two rings. The answer is no when he was wearing a watch. Then he took off the watch and said "now it is possible".