Zero Dimensional Space is T3

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

Then $T$ is a $T_3$ space.

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

Let $F \subseteq S$ be closed in $T$.

Then by definition of closed, $\complement_S \left({F}\right)$ is open in $T$, where $\complement_S \left({F}\right)$ is the complement of $F$ in $S$.

As $T$ is zero dimensional, it has a basis $\mathcal B$ which consists entirely of clopen sets.

As $\mathcal B$ is a basis for $T$, it follows that:
 * $\displaystyle \exists \mathcal U \subseteq \mathcal B: \complement_S \left({F}\right) = \bigcup \mathcal U$

that is, $\complement_S \left({F}\right)$ is the union of a subset of elements of $\mathcal B$.

But as all of $\mathcal U$ are clopen sets, then so is $\bigcup \mathcal U$.

Thus $F$ is also a clopen set, and therefore open in $T$.

So we have that $F$ and $\complement_S \left({F}\right)$ are open sets in $T$ such that:
 * $\exists U, V \in \vartheta: F \subseteq U, y \in V: U \cap V = \varnothing$

by setting $U = F$ and $V = \complement_S \left({F}\right)$.

That is, $T$ is a $T_3$ space.