Arens-Fort Space is not Locally Connected

Theorem
Let $T = \left({S, \tau}\right)$ be the Arens-Fort space.

Then $T$ is not a locally connected space.

Proof
Consider any open neighborhood $U$ of $\left({0, 0}\right)$

Consider $p \in S$ such that $p \notin U$.

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

However, as $U$ is open it also follows from the definition of the Arens-Fort space that $U \setminus \left\{{p}\right\}$ is also open in $T$.

So by definition of closed set, both $\left\{{p}\right\}$ and $\complement_S \left({\left\{{p}\right\}}\right)$ are clopen sets in $T$.

Hence the result, by definition of locally connected space.