# Arens-Fort Space is Completely Hausdorff

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## Theorem

Let $T = \struct {S, \tau}$ be the Arens-Fort space.

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

## Proof

We have:

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.

$\blacksquare$

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $26$. Arens-Fort Space: $1$