Perhaps everyone knows how to prove “there exist infinitely many primes” by one or more ways. Here I just want to share a proof using topology and prime factorization theorem. It is also (perhaps highly) possible that most of the readers know this proof except me, but I just think that it is interesting so I want to go through it here. Of course, the prerequisite is some point set topology.
Theorem. There exist infinitely many primes.
Proof. (by H. Furstenberg, 1955) Define a topology on by setting the basic open sets to be sets in the form
where is a nonzero integer and is an integer. This is a set of an arithmetic sequence. We call the collection . To show that this is a basis for some topology, we only need to show:
(2) The intersection of finitely many members of is in .
(1) is clear because . For (2), suppose is a basic set for each . The intersection is either empty, or contains some integer , and also
where is the lowest common multiple of . So (2) is justified and is a basis of some topology.
Notice that each is closed because its complement
is an open set.
By the prime factorization theorem, every integer other than 1 or -1 appears in some where is a positive prime. Also . Thus
which is a union of closed sets. If there are only finitely many primes, then the right hand side is a finite union of closed sets, which is closed. Then is closed so the set is open, which is impossible, because every open set is infinite, as each is so.