Totally Separated Space is Completely Hausdorff and Urysohn

Theorem
Let $T = \left({X, \vartheta}\right)$ be a topological space which is totally separated.

Then $T$ is completely Hausdorff and Urysohn.

Proof
Let $T = \left({X, \vartheta}\right)$ be a totally separated space.

Then for every $x, y \in X: x \ne y$ there exists a partition $U \mid V$ of $T$ such that $x \in U, y \in V$.

Having proved that $T$ is an Urysohn space, it follows from Urysohn Space is Completely Hausdorff Space that $T$ is also completely Hausdorff.