Discrete Space is Zero Dimensional/Proof 2

Proof
Let $\mathcal B$ be the set:
 * $\mathcal B := \left\{{\left\{{x}\right\}: x \in S}\right\}$

From Basis for Discrete Topology, $\mathcal B$ 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