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 . Arithmetic progressions themselves are by definition open, and in fact clopen, since

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

Consider the set , where the union runs over all primes . Then the complement of in is the single element , which is clearly not an open set (every open set is infinite in this topology). Thus is not closed, but since we have written as a union of closed sets and a finite 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.

Bibliography

1

Furstenberg, Harry, On the infinitude of primes, American Mathematical Monthly, Vol. 62, 1955, p. 353.

2

Ribenboim, Paulo. The New Book of Prime Number Records. Springer, 1996. p. 10