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: $1$