Adic Topologies for the Rational Integers

Abstract

AbstractA 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.