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Adic Topologies for the Rational Integers
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- Journal:
- Canadian Journal of Mathematics / Volume 55 / Issue 4 / 01 August 2003
- Published online by Cambridge University Press:
- 20 November 2018, pp. 711-723
- Print publication:
- 01 August 2003
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- Article
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- You have access
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A topology on
$\mathbb{Z}$, which gives a nice proof that the set of prime integers is infinite, is characterised and examined. It is found to be homeomorphic to 
$\mathbb{Q}$, with a compact completion homeomorphic to the Cantor set. It has a natural place in 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 the primes and squares, Fermat numbers, Fibonacci numbers and 
$k$-free numbers.
