Closed Set of Countable Fort Space is G-Delta

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Theorem

Let $T = \struct {S, \tau_p}$ 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:

$\DD = \set {S \setminus \set z: z \notin H}$

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

From its method of construction, $\DD$ has the same cardinality as $S \setminus H$ and so is countably infinite.

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

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


Let $r \in H$.

Then:

$\forall z \in S: z \notin H: r \in S \setminus \set z$

Thus by definition of $\DD$:

$\displaystyle H \subseteq \bigcap_{V \mathop \in \DD} V$


Let $r \notin H$.

Then:

$\exists z \in S: r \notin S \setminus \set z \in \DD$

So:

$\displaystyle \relcomp S H \subseteq \relcomp S {\bigcap_{V \mathop \in \DD} V}$

and so:

$\displaystyle H = \bigcap_{V \mathop \in \DD} V$

Hence the result.

$\blacksquare$