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 $q \ne (0,0)$.

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

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

If we take the neighborhoods of $(0,0)$ that are open we get a neighborhood basis $\mathcal{U}_p$ with the following property:


 * Since $(0,0) \in U \in \mathcal{U}_p$, its complement does not contain $(0,0)$ and so it is open.

This implies that $U$ is also closed.

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

This basis is formed with clopen sets.

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