Arens-Fort Space is Zero Dimensional

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

Then $T$ is zero dimensional.

Proof
Let $q \in S$ such that $q \ne \left({0, 0}\right)$.

Then from Clopen Points in Arens-Fort Space, $\left\{{q}\right\}$ is clopen.

So $\forall q \in S, q \ne \left({0, 0}\right): \left\{{\left\{{q}\right\}}\right\}$ is a local basis for $q$.

If we take the neighborhoods of $\left({0, 0}\right)$ that are open we get that they are also closed from Neighborhood of Origin of Arens-Fort Space is Closed.

The union of the local basis forms a basis for the topology.

This basis is formed with clopen sets.

So, by definition, $T$ is a zero dimensional space.