Countable Fort Space is Perfectly Normal

Theorem

Let $T = \struct {S, \tau_p}$ be a Fort space on a countably infinite set $S$.

Then $T$ is a perfectly normal space.

Proof

We have from Closed Set of Countable Fort Space is $G_\delta$ that every closed set in $T$ is a $G_\delta$ set.

From Fort Space is $T_5$ and $T_5$ Space is $T_4$ Space, we have that a Fort space is a $T_4$ space.

From Fort Space is $T_1$ it follows by definition that $T$ is a perfectly normal space.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $23$. Countable Fort Space: $3$