Adic Topologies for the Rational Integers

[摘要] A topology on $mathbb{Z}$, which gives a nice proof that theset of prime integers is infinite, is characterised and examined.It is found to be homeomorphic to $mathbb{Q}$, with a compactcompletion homeomorphic to the Cantor set. It has a natural placein a family of topologies on $mathbb{Z}$, which includes the$p$-adics, and one in which the set of rational primes $mathbb{P}$is dense. Examples from number theory are given, including theprimes and squares, Fermat numbers, Fibonacci numbers and $k$-freenumbers.

[学科分类] 数学（综合）