Existence of Hausdorff Space which is not Completely Hausdorff
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Theorem
There exists at least one example of a topological space which is a $T_2$ (Hausdorff) space, but is not also a completely Hausdorff space.
Proof
Let $T$ be an irrational slope topological space.
From Irrational Slope Space is $T_2$, $T$ is a $T_2$ (Hausdorff) space.
From Irrational Slope Space is not Completely Hausdorff, $T$ is not a completely Hausdorff 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 Regular Spaces