Urysohn Space is Completely Hausdorff Space

Theorem
Let $\struct {S, \tau}$ be an Urysohn space.

Then $\struct {S, \tau}$ is also a $T_{2 \frac 1 2}$ (completely Hausdorff) space.

Proof
Let $T = \struct {S, \tau}$ be an Urysohn space.

Then for any distinct points $x, y \in S$ (i.e. $x \ne y$), there exists an Urysohn function for $\set x$ and $\set y$.

Thus:
 * $\forall x, y \in S: x \ne y: \exists U, V \in \tau: x \in U, y \in V: U^- \cap V^- = \O$

which is precisely the definition of a $T_{2 \frac 1 2}$ (completely Hausdorff) space.