Completely Hausdorff space
In topology and related fields of mathematics, a completely Hausdorff space is a type of Hausdorff space satisfying a slightly stronger separation axiom.
Specifically, a topological space X is completely Hausdorff if any two distinct points can be separated by a function. That is, for any two points x, y in X there exists a continuous function f : X → R with f(x) = 0 and f(y) = 1.
This is a stronger condition than the Urysohn condition (points are separated by closed neighborhoods), which in turn, is stronger than the Hausdorff condition (points are separated by neighborhoods). Going in the other direction, every Tychonoff space is completely Hausdorff. In other words,
- Tychonoff ⇒ Completely Hausdorff ⇒ Urysohn ⇒ Hausdorff
N.B. Different sources give different definitions of a completely Hausdorff space. For example, Steen and Seebach exchange the meanings of comletely Hausdorff spaces and Urysohn spaces. 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. See History of separation axioms for more on this issue.
The property of being completely Hausdorff is a rather obscure requirement on a topological space. Most spaces studied in mathematics satisfy strictly weaker conditions (such as being Hausdorff), strictly stronger conditions (such as being Tychonoff), or unrelated conditions (such as being regular).
An example of a space which satisfies precisely this condition is the cocountable extension topology, which is the topology on the real line generated by the union of the usual Euclidean topology and the cocountable topology. Sets are open in this topology iff they are of the form U \ A where U is open in the Euclidean topology and A is countable. This space is completely Hausdorff, but not regular (and thus not Tychonoff). For more examples of spaces which satisfy precisely this condition see Steen and Seebach.
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