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:

[![enter image description here](https://i.stack.imgur.com/kv2Z9.png)](https://i.stack.imgur.com/kv2Z9.png)

**Edit:** The proof can also be found [here](https://proofwiki.org/wiki/Number_of_Primes_is_Infinite/Proof_2).

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?