# Arens-Fort Space is Completely Hausdorff

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## 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:

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

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{II}: \ 26: \ 1$