Discrete Space is Zero Dimensional

Theorem
Let $T = \left({S, \vartheta}\right)$ be a discrete space.

Then $T$ is zero dimensional.

Proof 1
We have from Partition of Singletons yields Discrete Topology that a discrete space is a partition space.

The result follows from Partition Topology is Zero Dimensional.

Proof 2
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 All Sets in Discrete Topology are Clopen, all the elements of $\mathcal B$ are both closed and open.

Hence the result, by definition of zero dimensional space