Arens-Fort Space is T5

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

Then $T$ is a $T_5$ space.


Let $A, B \subseteq S$ such that $A$ and $B$ are separated.

Let $p = \tuple {0, 0}$.

If $p \notin A$ and $p \notin B$ then $A$ and $B$ are both open and the problem is solved.

Otherwise $p$ must be in exactly one of them, because if $p$ were in both they could not be separated.

Without loss of generality, suppose $p \in A$.

Then $p \notin B$ so $B$ is open.

$B \cup \set p$ is closed by the definition of the Arens-Fort space.

From Set is Subset of its Topological Closure and Set Closure is Smallest Closed Set in Topological Space:

$B \subseteq B^- \subseteq B \cup \set p$

where $B^-$ is the closure of $B$.

So $B$ is closed or $p$ is in $B^-$.

But then $p \in A$, by hypothesis.

So $A \cap B^- \ne \O$ and so $A$ and $B$ are not separated.

So $B$ must be closed.

Therefore $\relcomp S B$ is an open set such that $A \subseteq \relcomp S B$, and the $T_5$ separation axiom is satisfied.