Fürstenberg’s proof of the infinitude of primes

Fürstenberg’s proof ([1], [2]) that there are infinitely many primes is an amusing and beautiful blend of elementary number theory and point-set topology.

Consider the arithmetic progression topology on the positive integers, where a basis of open sets is given by subsets of the form $U_{a,b}=\{n\in {\mathbb{Z}}^{+}|n\equiv bmoda\}$. Arithmetic progressions themselves are by definition open, and in fact clopen, since

$U_{a,b}^c={\displaystyle \bigcup _{c\ne b}}{U}_{a,c}$

where the union is taken over a set of distinct residue classes modulo $a$. Hence the complement of $U_{a,b}$ is a union of open sets and so is open, so $U_{a,b}$ itself is closed (and hence clopen).

Consider the set $U={\cup }_p{U}_{p,0}$, where the union runs over all primes $p$. Then the complement of $U$ in ${\mathbb{Z}}^{+}$ is the single element $\{1\}$, which is clearly not an open set (every open set is infinite in this topology). Thus $U$ is not closed, but since we have written $U$ as a union of closed sets and a union of closed sets is again closed, this implies that there must be infinitely many terms appearing in that union, i.e. that there must be infinitely many distinct primes.

Title	Fürstenberg’s proof of the infinitude of primes
Canonical name	FurstenbergsProofOfTheInfinitudeOfPrimes
Date of creation	2013-03-22 14:42:10
Last modified on	2013-03-22 14:42:10
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	8
Author	mathcam (2727)
Entry type	Proof
Classification	msc 11A41
Related topic	HausdorffSpaceNotCompletelyHausdorff