Arens-Fort Space is Completely Normal

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Theorem

Let $T = \left({S, \tau}\right)$ 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.


Sources