Definition:Appert Space
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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.
Also see
- Results about the Appert space can be found here.
Source of Name
This entry was named for Antoine Isidore Marie Joseph Appert.
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.): Part $\text {II}$: Counterexamples: $98$. Appert Space