\begin{definition}[Definition:Arens-Fort Space]
Let $S$ be the set $\Z_{\ge 0} \times \Z_{\ge 0}$ be the Cartesian product of the set of positive integers $\Z_{\ge 0}$:
:$S = \set {0, 1, 2, \ldots} \times \set {0, 1, 2, \ldots}$
Let $\tau \subseteq \powerset S$ be a subset of the power set of $S$ such that:
:$(1): \quad \forall H \subseteq S: \tuple {0, 0} \notin H \implies H \in \tau$
:$(2): \quad H \subseteq S: \tuple {0, 0} \in H$ and, for all but a finite number of $m \in \Z_{\ge 0}$, the sets $S_m$ defined as:
::::$S_m = \set {n: \tuple {m, n} \notin H}$
:::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 $\tuple {0, 0}$
:$H$ contains $\tuple {0, 0}$, 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 $T = \struct {S, \tau}$ is the '''Arens-Fort space'''.
\end{definition}