Urysohn Space is Completely Hausdorff Space

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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.

$\blacksquare$


Sources