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# Completely Hausdorff space

In mathematics, the class of completely Hausdorff topological spaces is a subclass of the Hausdorff spaces satisfying a stronger separation condition.

Different sources give different definitions of a completely Hausdorff space.

Counterexamples in Topology, by Lynn Arthur Steen and J. Arthur Seebach, Jr., defines completely Hausdorff to mean T, that is to say a topological space is completely Hausdorff if and only if any two distinct points are separated by closed neighbourhoods.

The Wikipedia article Separation axiom defines a topological space to be completely Hausdorff if and only if any two distinct points are separated by a continuous function. This is a stronger condition than T, since two points separated by a continuous function are necessarily separated by closed neighbourhoods. See that article for its sources.

The terms separated by closed neighbourhoods and separated by a continuous function are defined in the article Separated sets.

The study of separation axioms is notorious for conflicts of this kind. Readers of textbooks in topology must be sure to check the definitions used by the author.