One of the strange proofs (among the other beautiful proof) in the book "Proofs from the book" is the fifth one, which uses a special topology on the set of integer numbers, to prove there are infinite prime numbers.

My question is:

Is this method special just for this case, or is there anything deeper and this technique (or its generalization) can be used for some other kinds of problems?

A related question (by @ToddTrimble comment):

https://mathoverflow.net/questions/42589/is-fürstenbergs-topology-useful