Closed Set of Countable Fort Space is G-Delta

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Theorem

Let $T = \left({S, \tau_p}\right)$ be a Fort space on a countably infinite set $S$.

Let $H \subseteq S$ be closed in $T$.


Then $H$ is a $G_\delta$ set.


Proof

By definition of Fort space, $H$ is finite or contains $p$.

Consider the set of sets defined by $\mathcal D = \left\{{S \setminus \left\{{z}\right\}: z \notin H}\right\}$.

Because $H$ is finite, $S \setminus H$ is countably infinite.

From its method of construction, $\mathcal D$ has the same cardinality as $S \setminus H$ and so is countable.

We have that the elements of $\mathcal D$ are all open sets.

Thus, by definition, any set of the form $\displaystyle H \subseteq \bigcap_{V \mathop \in \mathcal D} V$ is a $G_\delta$ set.


Let $r \in H$.

Then:

$\forall z \in S: z \notin H: r \in S \setminus \left\{{z}\right\}$

Thus by definition of $\mathcal D$:

$\displaystyle H \subseteq \bigcap_{V \mathop \in \mathcal D} V$


Let $r \notin H$.

Then:

$\exists z \in S: r \notin S \setminus \left\{{z}\right\} \in \mathcal D$

So:

$\displaystyle \complement_S \left({H}\right) \subseteq \complement_S \left({\bigcap_{V \mathop \in \mathcal D} V}\right)$

and so:

$\displaystyle H = \bigcap_{V \mathop \in \mathcal D} V$

Hence the result.

$\blacksquare$