Arens-Fort Space is Totally Separated

Theorem

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

Then $T$ is totally separated.

Proof

We have that:

The Arens-Fort Space is Zero Dimensional.

The Arens-Fort Space is $T_1$ and so by $T_1$ Space is $T_0$ Space is a $T_0$ (Kolmogorov) space.

Then we have that a Zero Dimensional $T_0$ Space is Totally Separated.

$\blacksquare$

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: $8$