# Fürstenberg's Proof of the Infinitude of Primes

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.∎

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