Zero Dimensional T0 Space is Totally Separated

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

Then $T$ is totally separated.

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

As $T$ is zero dimensional, there exists a basis $\BB$ 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 that $\exists U \in \tau: x \in U, y \notin U$.

Then:
 * $\exists V \in \BB: x \in V$ and $V \subseteq U$

by definition of basis.

The set $V$ is clopen by the definition of $\BB$.

But then $x \in V$ which is open and $y \in S \setminus V$ which is also open.

$\set {V \mid S \setminus V}$ is a partition and hence $T$ is totally separated.