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

Let $p \ne \left({0, 0}\right)$ be a point in $U$.

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

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

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

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

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