In this note we would like to offer an elementary 'topological'
proof of the infinitude of the prime numbers. We introduce a
topology into the space of integers S, by using the arithmetic
progressions (from infinity to +infinity) as a basis. It is not
difficult to verify that this actually yields a topological space.
In fact, under this topology, S may be shown to be normal and hence
metrisable. Each arithmetic progression is closed as well as open,
since its complement is the union of the other arithmetic
progressions (having the same difference). As a result, the union
of any finite number of arithmetic progressions is closed.
Consider now the set A which is the union of A(p), where A(p)
consists of primes greater or equal to p. The only numbers not
belonging to A are 1 and 1, and since the set {1,1} is clearly
not an open set, A cannot be closed. Hence A is not a finite union
of closed sets, which proves that there is an infinity of primes.
 H. Fuerstenberg, On the infinitude of primes,
American Mathematical Montly, 62, 1955, p. 353
