Definition:Prime Integer Topology

Definition
Let $\Z_{>0}$ denote the set of (strictly) positive integers.

Let $\struct {\Z_{>0}, \tau}$ denote the relatively prime integer topology.

Let $\BB$ be the basis of $\struct {\Z_{>0}, \tau}$ of the form $\set {\map {U_a} b: a, b \in \Z_{>0} }$ where:
 * $\map {U_a} b = \set {b + n a \in \Z_{>0}: \gcd \set {a, b} = 1}$

where $\gcd \set {a, b}$ denotes the greatest common divisor of $a$ and $b$.

Let $\sigma$ be the subtopology of $\tau$ generated by the sub-basis $\PP$ of $\BB$ defined as:
 * $\PP := \set {\map {U_p} b: \text {$p$ is prime} }$

$\sigma$ is then referred to as the prime integer topology.

The topological space $T = \struct {\Z_{>0}, \sigma}$ is referred to as the prime integer space.

Also see

 * Prime Integer Topology is Topology