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 local base for the origin $\mathcal U_0$, and take $U \in \mathcal U_0$.

Let $p \ne (0,0)$ be a point in $U$.

From Clopen Points in Arens-Fort Space it follows that $\{p\}$ is clopen.

Since $p\in U$ and $(0,0)\in U$, $\{p\}$ is a clopen set contained in $U$ that $U\ne \{p\}$ because $(0,0)\in U\setminus\{p\}$.

So from Connected iff no Proper Clopen Sets, the set $U$ is not connected.

It is deduced that any local base is formed with disconnected sets.

Thus, by definition, $T$ is not locally connected.