Discrete Space is Zero Dimensional/Proof 2

Proof
Let $\BB$ be the set:
 * $\BB := \set {\set x: x \in S}$

From Basis for Discrete Topology, $\BB$ is a basis for $T$.

From Set in Discrete Topology is Clopen, all the elements of $\mathcal B$ are both closed and open.

Hence the result, by definition of zero dimensional space