Adic Topologies for the Rational Integers

A topology on Z, which gives a nice proof that the set of prime integers is infinite, is characterised and examined. It is found to be homeomorphic to Q, with a compact completion homeomorphic to the Cantor set. It has a natural place in a family of topologies on Z, which includes the p-adics, and one in which the set of rational primes P is dense. Examples from number theory are given, including the primes and squares, Fermat numbers, Fibonacci numbers and k-free numbers.