Definition:Arens-Fort Space

Definition
Let $S$ be the set $\Z_{\ge 0} \times \Z_{\ge 0}$ be the Cartesian product of the set of all positive integers:
 * $S = \left\{{0, 1, 2, \ldots}\right\} \times \left\{{0, 1, 2, \ldots}\right\}$

Let $\tau \subseteq \mathcal P \left({S}\right)$ be a subset of the power set of $S$ such that:
 * $(1): \quad \forall H \subseteq S: \left({0, 0}\right) \notin H \implies H \in \tau$


 * $(2): \quad H \subseteq S: \left({0, 0}\right) \in H$ and, for all but a finite number of $m \in \Z_{\ge 0}$, the sets $S_m$ defined as:
 * $S_m = \left\{{n: \left({m, n}\right) \notin H}\right\}$
 * are finite.

That is, $H$ is allowed to be in $\tau$ if, considering $S = \Z_{\ge 0} \times \Z_{\ge 0}$ as the lattice points of the first quadrant of a Cartesian plane:

Either:
 * $H$ does not contain $\left({0, 0}\right)$
 * $H$ contains $\left({0, 0}\right)$, and only a finite number of the columns of $S$ are allowed to omit an infinite number of points in $H$.

Then $\tau$ is the Arens-Fort topology on $S = \Z_{\ge 0} \times \Z_{\ge 0}$, and the topological space $T = \left({S, \tau}\right)$ is the Arens-Fort space.

Also see

 * Arens-Fort Topology is a Topology

It was created by Richard Friederich Arens as an adaption of a Fort Space.