Urysohn space - Revision history

Vipul: /* Relation with other properties */

2012-01-27T23:00:49Z

Relation with other properties

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 | [[Weaker than::normal Hausdorff space]] (also called <math>T_4</math>) || T1 and [[normal space|normal]] || || || {{intermediate notions short|Urysohn space|normal Hausdorff space}}
 |}

Vipul: /* Relation with other properties */

2012-01-27T22:47:30Z

Relation with other properties

← Older revision		Revision as of 22:47, 27 January 2012
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	==Relation with other properties==		==Relation with other properties==

			===Stronger properties===

	{\| class="sortable" border="1"		{\| class="sortable" border="1"

Vipul at 22:42, 27 January 2012

2012-01-27T22:42:29Z

← Older revision		Revision as of 22:42, 27 January 2012
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	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>.		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.		Note that the term Urysohn space is also used for the somewhat stronger notion of [[functionally Hausdorff space]]. There is a terminological ambiguity here.

	{{topospace property}}		{{topospace property}}

Vipul at 22:37, 27 January 2012

2012-01-27T22:37:15Z

← Older revision		Revision as of 22:37, 27 January 2012
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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}}
			\|}

Vipul: moved Urysohn space to Functionally Hausdorff space

2012-01-27T22:10:42Z

moved Urysohn space to Functionally Hausdorff space

New page

#REDIRECT [[Functionally Hausdorff space]]