Urysohn and completely Hausdorff spaces
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Separation axioms in topological spaces |
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|---|---|
| Kolmogorov classification | |
| T0 | (Kolmogorov) |
| T1 | (Fréchet) |
| T2 | (Hausdorff) |
| T2½ | (Urysohn) |
| completely T2 | (completely Hausdorff) |
| T3 | (regular Hausdorff) |
| T3½ | (Tychonoff) |
| T4 | (normal Hausdorff) |
| T5 | (completely normal Hausdorff) |
| T6 | (perfectly normal Hausdorff) |
| History | |
In topology, a discipline within mathematics, an Urysohn space, or T2½ space, is a topological space in which any two distinct points can be separated by closed neighborhoods. A completely Hausdorff space, or functionally Hausdorff space, is a topological space in which any two distinct points can be separated by a continuous function. These conditions are separation axioms that are somewhat stronger than the more familiar Hausdorff axiom T2.
Suppose that X is a topological space. Let x and y be points in X.
A Urysohn space, also called a T2½ space or Te space, is a space in which any two distinct points can be separated by closed neighborhoods.
A completely Hausdorff space, or functionally Hausdorff space, is a space in which any two distinct points can be separated by a continuous function.
The study of separation axioms is notorious for conflicts with naming conventions used. The definitions used in this article are those given by Willard (1970) and are the more modern definitions. Steen and Seebach (1970) and various other authors reverse the definition of completely Hausdorff spaces and Urysohn spaces. Readers of textbooks in topology must be sure to check the definitions used by the author. See History of the separation axioms for more on this issue.
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:
One can find counterexamples showing that none of these implications reverse.
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