Arens-Fort Space is not Connected

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

Then $T$ is not a connected space.

Proof
Consider $p \in S$ such that $p \ne \tuple {0, 0}$.

From Clopen Points in Arens-Fort Space, we have that $\set p$ is both open and closed in $T$.

So by definition of closed set, $\relcomp S {\set p}$ is also both open and closed in $T$.

So, by definition, $\set p \mid \relcomp S {\set p}$ is a separation of $T$

Hence the result, by definition of connected space.