Fürstenberg's Proof of the Infinitude of Primes

By Chris Caldwell

Euclid may have been the first to give a proof that there are infintely many primes.  Since then there have been many other proofs given.  Perhaps the strangest is the following topological proof by Fürstenberg [Fürstenberg55].  See the page "There are Infinitely Many Primes" for several other proofs.

Theorem.

There are infinitely many primes.

Proof.

Define a topology on the set of integers by using the arithmetic progressions (from -infinity to +infinity) as a basis.  It is easy to verify that this yields a topological space.  For each prime p let A_p consists of all multiples of p.  A_p is closed since its complement is the union of all the other arithmetic progressions with difference p.  Now let A be the union of the progressions A_p.  If the number of primes is finite, then A is a finite union of closed sets, hence closed.  But all integers except -1 and 1 are multiples of some prime, so the complement of A is {-1, 1} which is obviously not open.  This shows A is not a finite union and there are infinitely many primes.∎