H. Fürstenberg published in 1955 this proof that there are infinitely many prime numbers. It uses (of all things) *topology*, and is probably one of the most novel proofs of this fact. Furstenberg's work often involves using objects of one mathematical field in a very different one; this proof is an almost elementary example of this tendency. And it's almost completely unlike any other proof that there is no largest prime number you may have seen.

Define a topology on the integers **Z** by defining the open neighborhoods to be all arithmetic sequences S_{a,b}={an+b: n∈**Z**}. So a set B⊆**Z** is *open* iff it is empty or every x∈B has some a for which S_{a,x}⊆B.

This does actually define a topology on **Z**:

- ∅ and
**Z** are open sets.
- Any union of open sets is trivially open.
- The intersection of 2 open sets B
_{1},B_{2} is open. Indeed, if x∈B_{1}∩B_{2}, then there exist a_{1},a_{2} such that S_{ai,x}⊆B_{i} (i=1,2). Let a=lcm(a_{1},a_{2}) be the least common multiple of a_{1},a_{2} (or take a=a_{1}a_{2} if you prefer to be "wasteful" but get an even simpler proof). Then a gives a subsequence of our arithmetical progressions: S_{a,x}⊆S_{ai,x}. So x∈S_{a,x}⊆B_{1}∩B_{2}, and any finite intersection of open sets is open.

We only need 2 notes on this topology:

- Note that any non-empty open set contains an arithmetical progressions, hence it must be infinite.
- And note that any progressions S
_{a,b} is not only an open set but also a closed set, because S_{a,b}=**Z**\(S_{a,b+1}∪S_{a,b+2}∪...∪S_{a,b+a-1}) gives it as the complement of an open set.

Now any number apart from -1 and +1 appears in some arithmetical progression S

_{p,0} for some prime p. So

**Z**\{-1,+1} = ∪_{p prime}S_{p,0}.

By note "A", {-1,+1} isn't open, so the

LHS isn't a closed set. But if there were only finitely many primes, then the

RHS would be a finite union of closed (by note "B") sets, hence closed. Thus, there must be infinitely many primes.

An interesting exercise is to try to determine just *how* primes and divisibility are so important here. My take is that we only use multiplication when proving that the arithmetical sequences actually do give a topology...