In topology, a discipline within mathematics, an Urysohn space, or T_2½ space, is a topological space in which any two distinct points can be

separated by closed neighborhoods In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way: roughly speaking, neither overlapping nor touching. The notion of when two sets ...

. 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 In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...

. These conditions are separation axioms that are somewhat stronger than the more familiar Hausdorff axiom T₂.

Definitions

Suppose that ''X'' is a topological space. Let ''x'' and ''y'' be points in ''X''. *We say that ''x'' and ''y'' can be ''

'' if there exists a

closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...

neighborhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...

''U'' of ''x'' and a closed neighborhood ''V'' of ''y'' such that ''U'' and ''V'' are disjoint (''U'' ∩ ''V'' = ∅). (Note that a "closed neighborhood of ''x''" is a

closed set In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a cl ...

that contains an open set containing ''x''.) *We say that ''x'' and ''y'' can be '' separated by a function'' if there exists a

''f'' : ''X'' → ,1(the unit interval) with ''f''(''x'') = 0 and ''f''(''y'') = 1. A Urysohn space, also called a T_2½ 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.

Naming conventions

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.

Relation to other separation axioms

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 In topology and related fields of mathematics, a topological space ''X'' is called a regular space if every closed subset ''C'' of ''X'' and a point ''p'' not contained in ''C'' admit non-overlapping open neighborhoods. Thus ''p'' and ''C'' can b ...

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.

Examples

The cocountable extension topology is the topology on the

real line In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...

generated by the union of the usual Euclidean topology and the cocountable topology. Sets are open in this topology if and only if they are of the form ''U'' \ ''A'' where ''U'' is open in the Euclidean topology and ''A'' is countable. This space is completely Hausdorff and Urysohn, but not regular (and thus not Tychonoff). There exist spaces which are Hausdorff but not Urysohn, and spaces which are Urysohn but not completely Hausdorff or regular Hausdorff. Examples are non trivial; for details see Steen and Seebach.

Notes

References

* * Stephen Willard, ''General Topology'', Addison-Wesley, 1970. Reprinted by Dover Publications, New York, 2004. (Dover edition). * * {{planetmath reference, urlname=CompletelyHausdorff, title=Completely Hausdorff Separation axioms