Completely Hausdorff Space is Hausdorff Space

Theorem
Let $\struct {S, \tau}$ be a $T_{2 \frac 1 2}$ (completely Hausdorff) space.

Then $\struct {S, \tau}$ is also a $T_2$ (Hausdorff) space.

Proof
Let $T = \struct {S, \tau}$ be a completely Hausdorff space.

From the definition:


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

We have that Set is Subset of its Topological Closure and so $U \subseteq U^-$ and $V \subseteq V^-$.

This leads to:


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

which is precisely the definition of a Hausdorff ($T_2$) space.