Arens-Fort Space is Expansion of Countable Fort Space

Theorem

Let $T = \struct {S, \tau}$ be the Arens-Fort space, where $S = \Z_{\ge 0} \times \Z_{\ge 0}$.

Let $T_p = \struct {S, \tau_p}$ be the Fort space on $S$ where $p = \left({0, 0}\right)$.

Then $\tau$ is an expansion of $\tau_p$.

Let $H \in \tau_p$ where $p = \tuple {0, 0}$.

Then either:

$(1): \quad \tuple {0, 0} \in \relcomp S H$

or:

$(2): \quad H$ is cofinite in $S$, that is $\relcomp S H$ is finite.

Case $(1)$ means that $\tuple {0, 0} \notin H$ and so $H \in \tau$ by definition of the Arens-Fort space.

Suppose case $(2)$ applies.

Then for all $m \in \Z_{\ge 0}$, the sets $S_m$ defined as $S_m = \set {n: \tuple {m, n} \notin H}$ are finite.

Thus in a finite number of these (that is, none of them) the set $S_m$ is infinite.

So $H \in \tau$.

So, if $H \in \tau_p$ it follows that $H \in \tau$ and so $\tau_p \subseteq \tau$.

Hence the result by definition of expansion.

$\blacksquare$