Completely Hausdorff Space - Relation To Other Separation Axioms

Relation To Other Separation Axioms

It is an easy exercise to show that any two points which can be separated by a function can be separated by closed neighborhoods. If they can be separated by closed neighborhoods then clearly they can be separated by neighborhoods. It follows that every completely Hausdorff space is Urysohn and every Urysohn space is Hausdorff.

One can also show that every regular Hausdorff space is Urysohn and every Tychonoff space (=completely regular Hausdorff space) is completely Hausdorff. In summary we have the following implications:

Tychonoff (T) regular Hausdorff (T3)
completely Hausdorff Urysohn (T) Hausdorff (T2) T1

One can find counterexamples showing that none of these implications reverse.

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