Definition:Prime Integer Topology
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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
- Results about the prime integer topology can be found here.
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (next): Part $\text {II}$: Counterexamples: $61$. Prime Integer Topology