I was reading Proofs from THE BOOK by Martin Aigner, Günter M. Ziegler and was very impressed by the following proof of infinity of prime numbers with topology:
Edit: The proof can also be found here.
Though I understand every step of this proof logically, I still find using topology to prove this classical result in Number Theory amazing. I wonder why there is a topological proof to the infinity of prime numbers. What is the intuition that motives this proof? Are there any topological proofs to other famous results in other fields of mathematics?