relatively prime integer topology

Let $X$ be the set of strictly positive integers. The relatively prime integer topology on $X$ is the topology determined by a basis consisting of the sets

$U(a,b)=\{ax+b\mid x\in X\}$

for any $a$ and $b$ are relatively prime integers. That this does indeed form a basis is found in this entry. (http://planetmath.org/HausdorffSpaceNotCompletelyHausdorff)

Equipped with this topology, $X$ is $T_0$ (http://planetmath.org/T0Space), $T_1$ (http://planetmath.org/T1Space),and $T_2$ (http://planetmath.org/T2Space), but satisfies none of the higher separation axioms (and hence meet very few compactness criteria).

We can define a coarser topology on $X$ by considering the subbasis of the above basis consisting of all $U(a,b)$ with $a$ being a prime. This is called the prime integer topology on ${\mathbb{Z}}^{+}$.