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.