View source for Urysohn and completely Hausdorff spaces

{{Short description|Form of topological spaces}}
{{redirect-confuse|Urysohn space|Urysohn universal space}}
{{Separation axioms}}

In [[topology]], a discipline within mathematics, an '''Urysohn space''', or '''T<sub>2½</sub> 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 axiom]]s that are somewhat stronger than the more familiar [[Hausdorff space|Hausdorff axiom]] T<sub>2</sub>.

==Definitions==

Suppose that ''X'' is a [[topological space]]. Let ''x'' and ''y'' be points in ''X''.
*We say that ''x'' and ''y'' can be ''[[separated by closed neighborhoods]]'' if there exists a [[closed set|closed]] [[neighborhood (topology)|neighborhood]] ''U'' of ''x'' and a closed neighborhood ''V'' of ''y'' such that ''U'' and ''V'' are [[disjoint sets|disjoint]] (''U'' ∩ ''V'' = ∅). (Note that a "closed neighborhood of ''x''" is a [[closed set]] that contains an [[open set]] containing ''x''.)
*We say that ''x'' and ''y'' can be ''[[separated by a function]]'' if there exists a [[continuity (topology)|continuous function]] ''f'' : ''X'' → [0,1] (the [[unit interval]]) with ''f''(''x'') = 0 and ''f''(''y'') = 1.

A '''Urysohn space''', also called a '''T<sub>2½</sub> 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 space|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:
:{| style="text-align: center;"
|-
| [[Tychonoff space|Tychonoff]] (T<sub>3½</sub>) || &nbsp;<math>\Rightarrow</math>&nbsp;
| [[regular space|regular Hausdorff]] (T<sub>3</sub>) 
|-
|<math>\Downarrow</math> || || <math>\Downarrow</math>
|-
| completely Hausdorff || &nbsp;<math>\Rightarrow</math>&nbsp;
| Urysohn (T<sub>2½</sub>) || &nbsp;<math>\Rightarrow</math>&nbsp;
| [[Hausdorff space|Hausdorff]] (T<sub>2</sub>) || &nbsp;<math>\Rightarrow</math>&nbsp;
| [[T1 space|T<sub>1</sub>]]
|}
One can find counterexamples showing that none of these implications reverse.<ref>{{planetmath reference|urlname=HausdorffSpaceNotCompletelyHausdorff|title=Hausdorff space not completely Hausdorff}}</ref>

==Examples==

The [[cocountable extension topology]] is the topology on the [[real line]] generated by the [[union (set theory)|union]] of the usual [[Euclidean topology]] and the [[cocountable topology]]. Sets are [[open set|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/>

== References ==
* {{Citation | last1=Steen | first1=Lynn Arthur | author1-link=Lynn Arthur Steen | last2=Seebach | first2=J. Arthur Jr. | author2-link=J. Arthur Seebach, Jr. | title=[[Counterexamples in Topology]] | orig-year=1978 | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=[[Dover Publications|Dover]] reprint of 1978 | isbn=978-0-486-68735-3 | mr=507446 | year=1995}}
* Stephen Willard, ''General Topology'', Addison-Wesley, 1970. Reprinted by Dover Publications, New York, 2004. {{isbn|0-486-43479-6}} (Dover edition).
* {{Willard General Topology}} <!-- {{sfn|Willard|2004|p=}} -->
* {{planetmath reference|urlname=CompletelyHausdorff|title=Completely Hausdorff}}

[[Category:Separation axioms]]