Arens-Fort Space is Expansion of Countable Fort Space

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Theorem

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

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


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


Furthermore, $S$ is countably infinite, so $T_p$ is a countable Fort space.


Proof

Let $H \in \tau_p$ where $p = \left({0, 0}\right)$.

Then either:

$(1): \quad \left({0, 0}\right) \in \complement_S \left({H}\right)$

or:

$(2): \quad H$ is cofinite in $S$, i.e. $\complement_S \left({H}\right)$ is finite.


Case $(1)$ means that $\left({0, 0}\right) \notin H$ 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 $S_m = \left\{{n: \left({m, n}\right) \notin H}\right\}$ 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.


Finally, from Arens-Fort Space is Countable we have that $S$ is countably infinite.

$\blacksquare$


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