Interesting application of topology to another math area

Question

I'm looking for some surprising theorem that links topology and other branch of math. Any help would be great, of course. However, I'd say that some areas are more attractive to me. In decreasing order, they are:

• Game theory
• Functional analysis
• Measure theory
• Abstract algebra
• ...

I wouldn't like to explore areas as Probability, Graph theory, Number theory and ODE.

Furthermore, I'd like to find something more complex than the proofs of The fundamental theorem of Algebra and Brouwer's fixed-point theorem using Algebraic Topology, for instance.

Here we are presented the Uniform boundedness principle of Functional analysis as a very important application of the Baire Category Theorem. It was the "really simple elementary proof" without Baire what makes me give it up.

I'd be grateful with a hint. Thanks in advance!

Answer 1

There's the Zariski topology, in which the set of prime ideals in a commutative ring is made into a topological space (with the ideals as points), so any of the many theorems involving this notion serves as an example that connects commutative ring theory and general topology.