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.

$\blacksquare$


Sources

  • 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $2$: Separation Axioms: Completely Hausdorff Spaces
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