Definition:Appert Space

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

For a given subset $H$ of $\Z_{>0}$ and element $n$ of $\Z_{>0}$, let $\map N {n, H}$ denote the number of integers in $E$ which are less than or equal to $n$:


 * $\forall n \in \Z_{>0}, H \subseteq \Z_{>0}: \map N {n, H} = \card {\set {x \in H: x \le n} }$

Let $\tau$ be the topology defined as:


 * $1 \notin H \implies H \in \tau$


 * $1 \in H$ and $\ds \lim_{n \mathop \to \infty} \dfrac {\map N {n, H} } n = 1 \implies H \in \tau$

for $H \subseteq \Z_{>0}$.

$\tau$ is referred to as the Appert topology.

The topological space $T = \struct {S, \tau}$ is referred to as the Appert space.