Zero Dimensional T0 Space is Totally Separated

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

Then $T$ is totally separated.

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

As $T$ is zero dimensional, there exists a basis $\mathcal B$ whose sets are all clopen.

Let $x, y \in S$.

As $T$ is a $T_0$ space:
 * $\exists U \in \tau: x \in U, y \notin U$

or:
 * $\exists U \in \tau: y \in U, x \notin U$

Suppose WLOG that $\exists U \in \tau: x \in U, y \notin U$.

Then:
 * $\displaystyle \exists \mathcal U \subseteq \mathcal B: U = \bigcup \mathcal U$

by definition of basis.

But then $U$ is also open as it is the union of a set of open sets of $T$.