# 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 positive integers $\Z$:

$S = \set {0, 1, 2, \ldots} \times \set {0, 1, 2, \ldots}$

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.

## Also see

• Results about the Arens-Fort space can be found here.

## Source of Name

This entry was named for Richard Friederich Arens and Marion Kirkland Fort, Jr.

## Historical Note

The Arens-Fort space was created by Richard Friederich Arens as an adaption of a Fort Space.