Arens-Fort Space is Completely Hausdorff

Theorem

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

Then $T$ is a $T_{2 \frac 1 2}$ (completely Hausdorff) space.

Proof

We have:

Fort Space is Completely Normal
Arens-Fort Space is Expansion of Fort Space.

From Sequence of Implications of Separation Axioms we have that a Fort space is a $T_{2 \frac 1 2}$ space.

Then in Separation Properties Preserved by Expansion, it follows that as a Fort space is a $T_{2 \frac 1 2}$ space, then so is the Arens-Fort space.

$\blacksquare$