Arens-Fort Space is T1

Theorem

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

Then $T$ is a $T_1$ (Fréchet) space.

Proof

We have that the Arens-Fort Space is Completely Hausdorff.

Then from Sequence of Implications of Separation Axioms, a $T_{2 \frac 1 2}$ (completely Hausdorff) space is a $T_1$ 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: $1$