topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...

, 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. 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 neighborhood ''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 spac ...

that contains an

open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that a ...

containing ''x''.) *We say that ''x'' and ''y'' can be '' separated by a function'' if there exists a continuous function ''f'' : ''X'' → ,1(the

unit interval In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analys ...

) 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 The history of the separation axioms in general topology has been convoluted, with many meanings competing for the same terms and many terms competing for the same concept. Origins Before the current general definition of topological space, ...

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 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 number to a po ...

generated by the union of the usual

Euclidean topology In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean metric. Definition The Euclidean norm on \R^n is the non-negative function \, \cdo ...

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