Zero Dimensional T0 Space is Totally Separated

Theorem
Let $T = \left({X, \vartheta}\right)$ be a zero dimensional topological space which is also a $T_0$ (Kolmogorov) space.

Then $T$ is totally separated.

Proof
Let $T = \left({X, \vartheta}\right)$ be a zero dimensional space which is also a $T_0$ (Kolmogorov) space.