# Arens-Fort Space is Completely Normal

## Theorem

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

Then $T$ is a completely normal space.

Consequently, $T$ satisfies all weaker separation axioms.

## Proof

We have:

Arens-Fort Space is $T_1$
Arens-Fort Space is $T_5$

and so by definition $T$ is completely normal.

$\blacksquare$

See Sequence of Implications of Separation Axioms for confirmation of the statement about weaker separation axioms.