Urysohn space: Difference between revisions

Revision as of 22:37, 27 January 2012

Definition

A topological space $X$ is termed a Urysohn space if, for any two distinct points $x, y \in X$ , there exist disjoint open subsets $U ∋ x, V ∋ y$ such that the closures <math\overline{U}</math> and $\bar{V}$ are disjoint closed subsets of $X$ .

Note that the term Urysohn space is also used for the somewhat stronger notion of functinally Hausdorff space. There is a terminological ambiguity here.

This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces

In the T family (properties of topological spaces related to separation axioms), this is called: T2.5

Relation with other properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
regular Hausdorff space (also called $T_{3}$ )	T1 and any point can be separated from any disjoint closed subset	regular Hausdorff implies Urysohn	Urysohn not implies regular	\|FULL LIST, MORE INFO
functionally Hausdorff space				\|FULL LIST, MORE INFO
Tychonoff space (also called $T_{3.5}$ )	T1 and any point and disjoint closed subset can be separated by a continuous function			Functionally Hausdorff space\|FULL LIST, MORE INFO
normal Hausdorff space (also called $T_{4}$ )	T1 and normal			Functionally Hausdorff space\|FULL LIST, MORE INFO

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-#REDIRECT [[Functionally Hausdorff space]]
+==Definition==
+A [[topological space]] <math>X</math> is termed a '''Urysohn space''' if, for any two ''distinct'' points <math>x,y \in X</matH>, there exist disjoint [[open subset]]s <math>U \ni x, V \ni y </math> such that the [[closure]]s <math\overline{U}</math> and <math>\overline{V}</math> are disjoint [[closed subset]]s of <math>X</math>.
+Note that the term Urysohn space is also used for the somewhat stronger notion of [[functinally Hausdorff space]]. There is a terminological ambiguity here.
+{{topospace property}}
+{{T family|T2.5}}
+==Relation with other properties==
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Weaker than::regular Hausdorff space]] (also called <math>T_3</math>) || T1 and any point can be separated from any disjoint closed subset || [[regular Hausdorff implies Urysohn]] || [[Urysohn not implies regular]] || {{intermediate notions short|Urysohn space|regular Hausdorff space}}
+|-
+| [[Weaker than::functionally Hausdorff space]] || || || || {{intermediate notions short|Urysohn space|functionally Hausdorff space}}
+|-
+| [[Weaker than::Tychonoff space]] (also called <math>T_{3.5}</math>) || T1 and any point and disjoint closed subset can be separated by a continuous function || || ||  {{intermediate notions short|Urysohn space|Tychonoff space}}
+|-
+| [[Weaker than::normal Hausdorff space]] (also called <math>T_4</math>) || T1 and [[normal space|normal]] || || || {{intermediate notions short|Urysohn space|normal Hausdorff space}}
+|}