Functionally Hausdorff space - Revision history

Vipul: /* Metaproperties */

2012-01-27T23:14:12Z

Metaproperties

← Older revision		Revision as of 23:14, 27 January 2012
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	!Metaproperty name !! Satisfied? !! Proof		!Metaproperty name !! Satisfied? !! Proof
	\|-		\|-
	\|[[satisfies metaproperty::subspace-hereditary property of topological spaces]] \|\| Yes \|\| [[~~Urysohn~~ is hereditary]]		\|[[satisfies metaproperty::subspace-hereditary property of topological spaces]] \|\| Yes \|\| [[functional Hausdorffness is hereditary]]
	\|-		\|-
	\|[[satisfies metaproperty::refining-preserved property of topological spaces]] \|\| Yes \|\| [[~~Urysohn~~ is refining-preserved]]		\|[[satisfies metaproperty::refining-preserved property of topological spaces]] \|\| Yes \|\| [[functional Hausdorffness is refining-preserved]]
	\|-		\|-
	\|[[satisfies metaproperty::product-closed property of topological spaces]]\|\| Yes \|\| [[~~Urysohn~~ is product-closed]]		\|[[satisfies metaproperty::product-closed property of topological spaces]]\|\| Yes \|\| [[functional Hausdorffness is product-closed]]
	\|-		\|-
	\|[[satisfies metaproperty::box product-closed property of topological spaces]] \|\| Yes \|\| [[~~Urysohn~~ is box product-closed]]		\|[[satisfies metaproperty::box product-closed property of topological spaces]] \|\| Yes \|\| [[functional Hausdorffness is box product-closed]]
	\|}		\|}

Vipul at 22:16, 27 January 2012

2012-01-27T22:16:58Z

← Older revision		Revision as of 22:16, 27 January 2012
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	==Definition==		==Definition==

	A [[topological space]] is termed a '''completely Hausdorff space''' or '''~~Urysohn~~ space''' if it satisfies the following equivalent conditions:		A [[topological space]] is termed a '''completely Hausdorff space''' or '''functionally Hausdorff space''' if it satisfies the following equivalent conditions:

	# For any two points in it, there is a continuous function from the whole space to <math>[0,1]</math> that takes the value <math>0</math> at one point and <math>1</math> at the other.		# For any two points in it, there is a continuous function from the whole space to <math>[0,1]</math> that takes the value <math>0</math> at one point and <math>1</math> at the other.
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	{{variationof\|Hausdorffness}}		{{variationof\|Hausdorffness}}

	~~{{T family\|T2.5}}~~
	==Relation with other properties==		==Relation with other properties==

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	! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	\|-		\|-
	\| [[Weaker than::normal space]] \|\| <math>T_1</math> and any two disjoint closed subsets are separated by disjoint open subsets \|\| [[normal implies ~~Urysohn~~]] \|\| [[~~Urysohn~~ not implies normal]] \|\| {{intermediate notions short\|~~Urysohn~~ space\|normal space}}		\| [[Weaker than::normal Hausdorff space]] \|\| <math>T_1</math> and any two disjoint closed subsets are separated by disjoint open subsets \|\| [[normal Hausdorff implies functionally Hausdorff]] \|\| [[functionally Hausdorff not implies normal]] \|\| {{intermediate notions short\|functionally Hausdorff space\|normal Hausdorff space}}
	\|-		\|-
	\| [[Weaker than::~~completely regular~~ space]] \|\| <math>T_1</math> and continuous function to <math>[0,1]</math> separating any point and disjoint closed subset \|\| [[~~completely regular~~ implies ~~Urysohn~~]] \|\| [[~~Urysohn~~ not implies completely regular]] \|\| {{intermediate notions short\|~~Urysohn~~ space\|~~completely regular~~ space}}		\| [[Weaker than::Tychonoff space]] \|\| <math>T_1</math> and [[completely regular space\|completely regular]]: continuous function to <math>[0,1]</math> separating any point and disjoint closed subset \|\| [[Tychonoff implies functionally Hausdorff]] \|\| [[functionally Hausdorff not implies completely regular]] \|\| {{intermediate notions short\|functionally Hausdorff space\|Tychonoff space}}
	\|-		\|-
	\| [[Weaker than::metrizable space]] \|\| \|\| \|\| \|\| {{intermediate notions short\|~~Urysohn~~ space\|metrizable space}}		\| [[Weaker than::metrizable space]] \|\| \|\| \|\| \|\| {{intermediate notions short\|functionally Hausdorff space\|metrizable space}}
	\|-		\|-
	\| [[Weaker than::submetrizable space]] \|\| either metrizable or can become metrizable upon passing to a coarser topology \|\| \|\| \|\| {{intermediate notions short\|~~Urysohn~~ space\|submetrizable space}}		\| [[Weaker than::submetrizable space]] \|\| either metrizable or can become metrizable upon passing to a coarser topology \|\| \|\| \|\| {{intermediate notions short\|functionally Hausdorff space\|submetrizable space}}
	\|}		\|}

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	! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	\|-		\|-
	\| [[Stronger than::~~Hausdorff~~ space]] \|\| distinct points separated by disjoint open subsets \|\| [[Urysohn ~~implies~~ Hausdorff]] \|\| [[Hausdorff ~~not implies Urysohn~~]] \|\| {{intermediate notions short\|Hausdorff space\|~~Urysohn~~ space}}		\| [[Stronger than::Urysohn space]] \|\| any two distinct points can be separated by disjoint open subsets whose closures are also disjoint \|\| \|\| \|\| {{intermediate notions short\|Urysohn space\|functionally Hausdorff space}}
			\|-
			\| [[Stronger than::Hausdorff space]] \|\| distinct points separated by disjoint open subsets \|\| \|\| \|\| {{intermediate notions short\|Hausdorff space\|functionally Hausdorff space}}
	\|-		\|-
	\| [[Stronger than::T1 space]] \|\| points are closed \|\| (via Hausdorff) \|\| (via Hausdorff) \|\| {{intermediate notions short\|T1 space\|Urysohn space}}		\| [[Stronger than::T1 space]] \|\| points are closed \|\| (via Hausdorff) \|\| (via Hausdorff) \|\| {{intermediate notions short\|T1 space\|Urysohn space}}

Vipul: moved Urysohn space to Functionally Hausdorff space

2012-01-27T22:10:42Z

moved Urysohn space to Functionally Hausdorff space

← Older revision	Revision as of 22:10, 27 January 2012
(No difference)

Vipul: /* Relation with other properties */

2012-01-25T02:35:17Z

Relation with other properties

← Older revision		Revision as of 02:35, 25 January 2012
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	\|-		\|-
	\| [[Weaker than::completely regular space]] \|\| <math>T_1</math> and continuous function to <math>[0,1]</math> separating any point and disjoint closed subset \|\| [[completely regular implies Urysohn]] \|\| [[Urysohn not implies completely regular]] \|\| {{intermediate notions short\|Urysohn space\|completely regular space}}		\| [[Weaker than::completely regular space]] \|\| <math>T_1</math> and continuous function to <math>[0,1]</math> separating any point and disjoint closed subset \|\| [[completely regular implies Urysohn]] \|\| [[Urysohn not implies completely regular]] \|\| {{intermediate notions short\|Urysohn space\|completely regular space}}
			\|-
			\| [[Weaker than::metrizable space]] \|\| \|\| \|\| \|\| {{intermediate notions short\|Urysohn space\|metrizable space}}
			\|-
			\| [[Weaker than::submetrizable space]] \|\| either metrizable or can become metrizable upon passing to a coarser topology \|\| \|\| \|\| {{intermediate notions short\|Urysohn space\|submetrizable space}}
	\|}		\|}

Vipul: /* Relation with other properties */

2012-01-24T23:03:07Z

Relation with other properties

← Older revision		Revision as of 23:03, 24 January 2012
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	\| [[Weaker than::normal space]] \|\| <math>T_1</math> and any two disjoint closed subsets are separated by disjoint open subsets \|\| [[normal implies Urysohn]] \|\| [[Urysohn not implies normal]] \|\| {{intermediate notions short\|Urysohn space\|normal space}}		\| [[Weaker than::normal space]] \|\| <math>T_1</math> and any two disjoint closed subsets are separated by disjoint open subsets \|\| [[normal implies Urysohn]] \|\| [[Urysohn not implies normal]] \|\| {{intermediate notions short\|Urysohn space\|normal space}}
	\|-		\|-
	\| [[Weaker than::~~nompletely~~ regular space]] \|\| <math>T_1</math> and continuous function to <math>[0,1]</math> separating any point and disjoint closed subset \|\| [[completely regular implies Urysohn]] \|\| [[Urysohn not implies completely regular]] \|\| {{intermediate notions short\|Urysohn space\|completely regular space}}		\| [[Weaker than::completely regular space]] \|\| <math>T_1</math> and continuous function to <math>[0,1]</math> separating any point and disjoint closed subset \|\| [[completely regular implies Urysohn]] \|\| [[Urysohn not implies completely regular]] \|\| {{intermediate notions short\|Urysohn space\|completely regular space}}
	\|}		\|}

Vipul at 22:35, 24 January 2012

2012-01-24T22:35:54Z

← Older revision		Revision as of 22:35, 24 January 2012
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			==Definition==

			A [[topological space]] is termed a '''completely Hausdorff space''' or '''Urysohn space''' if it satisfies the following equivalent conditions:

			# For any two points in it, there is a continuous function from the whole space to <math>[0,1]</math> that takes the value <math>0</math> at one point and <math>1</math> at the other.
			# For any two points in it, there is a continuous function from the whole space to the reals that takes the value <math>0</math> at one point and <math>1</math> at the other.
			# For any two points in it, there is a continuous function from the whole space to the reals that takes distinct values at the two points.
			# For any two points in it, and any two specified distinct real numbers, there is a continuous function from the whole space to the reals that takes the two specified values at the two points.

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

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	{{T family\|T2.5}}		{{T family\|T2.5}}

	~~==Definition==~~

	A [[topological space]] is termed a '''completely Hausdorff space''' or '''Urysohn space''' if for any two points in it, there is a continuous function from the whole space to <math>[0,1]</math> that takes the value <math>0</math> at one point and <math>1</math> at the other.

	==Relation with other properties==		==Relation with other properties==

	===Stronger properties===		===Stronger properties===

	{\| class="~~wikitable~~" border="1"		{\| class="sortable" border="1"
	! ~~property~~ !! ~~quick description~~ !! ~~proof~~ of implication !! ~~proof~~ of strictness (reverse implication failure) !! ~~intermediate~~ notions		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	\|-		\|-
	\| [[Weaker than::~~Normal~~ space]] \|\| <math>T_1</math> and any two disjoint closed subsets are separated by disjoint open subsets \|\| [[normal implies Urysohn]] \|\| [[Urysohn not implies normal]] \|\| {{intermediate notions short\|Urysohn space\|normal space}}		\| [[Weaker than::normal space]] \|\| <math>T_1</math> and any two disjoint closed subsets are separated by disjoint open subsets \|\| [[normal implies Urysohn]] \|\| [[Urysohn not implies normal]] \|\| {{intermediate notions short\|Urysohn space\|normal space}}
	\|-		\|-
	\| [[Weaker than::~~Completely~~ regular space]] \|\| <math>T_1</math> and continuous function to <math>[0,1]</math> separating any point and disjoint closed subset \|\| [[completely regular implies Urysohn]] \|\| [[Urysohn not implies completely regular]] \|\| {{intermediate notions short\|Urysohn space\|completely regular space}}		\| [[Weaker than::nompletely regular space]] \|\| <math>T_1</math> and continuous function to <math>[0,1]</math> separating any point and disjoint closed subset \|\| [[completely regular implies Urysohn]] \|\| [[Urysohn not implies completely regular]] \|\| {{intermediate notions short\|Urysohn space\|completely regular space}}
	\|}		\|}

	===Weaker properties===		===Weaker properties===

	{\| class="~~wikitable~~" border="1"		{\| class="sortable" border="1"
	! ~~property~~ !! ~~quick description~~ !! ~~proof~~ of implication !! ~~proof~~ of strictness (reverse implication failure) !! ~~intermediate~~ notions		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	\|-		\|-
	\| [[Stronger than::Hausdorff space]] \|\| distinct points separated by disjoint open subsets \|\| [[Urysohn implies Hausdorff]] \|\| [[Hausdorff not implies Urysohn]] \|\| {{intermediate notions short\|Hausdorff space\|Urysohn space}}		\| [[Stronger than::Hausdorff space]] \|\| distinct points separated by disjoint open subsets \|\| [[Urysohn implies Hausdorff]] \|\| [[Hausdorff not implies Urysohn]] \|\| {{intermediate notions short\|Hausdorff space\|Urysohn space}}
	\|-		\|-
	\| [[Stronger than::T1 space]] \|\| points are closed \|\| (via Hausdorff) \|\| (via Hausdorff) \|\| {{intermediate notions short\|T1 space\|Urysohn space}}		\| [[Stronger than::T1 space]] \|\| points are closed \|\| (via Hausdorff) \|\| (via Hausdorff) \|\| {{intermediate notions short\|T1 space\|Urysohn space}}
			\|-
			\|[[Stronger than::Kolmogorov space]] \|\| points are distinguishable \|\| (via Hausdorff) \|\| (via Hausdorff) \|\| {{intermediate notions short\|Kolmogorov space\|Urysohn space}}
	\|}		\|}

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	{\| class="wikitable" border="1"		{\| class="wikitable" border="1"
	!Metaproperty name !! Satisfied? !! Proof ~~!! Section in this article~~		!Metaproperty name !! Satisfied? !! Proof
	\|-		\|-
	\|[[satisfies metaproperty::subspace-hereditary property of topological spaces]] \|\| Yes \|\| [[Urysohn is hereditary~~]] \|\| [[#Hereditariness~~]]		\|[[satisfies metaproperty::subspace-hereditary property of topological spaces]] \|\| Yes \|\| [[Urysohn is hereditary]]
	\|-		\|-
	\|[[satisfies metaproperty::refining-preserved property of topological spaces]] \|\| Yes \|\| [[Urysohn is refining-preserved~~]] \|\| [[#Refining~~]]		\|[[satisfies metaproperty::refining-preserved property of topological spaces]] \|\| Yes \|\| [[Urysohn is refining-preserved]]
	\|-		\|-
	\|[[satisfies metaproperty::product-closed property of topological spaces]]\|\| Yes \|\| [[Urysohn is product-closed~~]] \|\| [[#Products~~]]		\|[[satisfies metaproperty::product-closed property of topological spaces]]\|\| Yes \|\| [[Urysohn is product-closed]]
	\|-		\|-
	\|[[satisfies metaproperty::box product-closed property of topological spaces]] \|\| Yes \|\| [[Urysohn is box product-closed~~]] \|\| [[#Box products~~]]		\|[[satisfies metaproperty::box product-closed property of topological spaces]] \|\| Yes \|\| [[Urysohn is box product-closed]]
	\|}		\|}

	~~{{subspace-closed}}~~

	~~Any subspace of a Urysohn space is Urysohn. {{proofat\|[[Urysohn is hereditary]]}}~~

	~~{{further\|[[Hausdorffness is hereditary]], [[T1 is hereditary]], [[regularity is hereditary]], [[complete regularity is hereditary]], [[normality is not hereditary]]}}~~

	~~{{refining-preserved}}~~

	If <math>X</math> is a Urysohn space with topology <math>\tau</math>, and <math>\tau'</math> is a finer topology than <math>\tau</math>, then <math>(X,\tau')</math> is also a Urysohn space. {{proofat\|[[Urysohn is refining-preserved]]}}

	~~{{DP-closed}}~~

	An arbitrary product of Urysohn spaces, equipped with the [[product topology]], is also a Urysohn space. We can define the continuous function by simply concentrating on one coordinate where the points differ. {{proofat\|[[Urysohn is product-closed]]}}

	~~{{further\|[[Hausdorffness is product-closed]]}}~~

	~~{{box product-closed}}~~

	~~An arbitrary box product of Urysohn spaces, equipped wit the [[box topology]], is also a Urysohn space. {{proofat\|[[Urysohn is box product-closed]]}}~~

Vipul at 19:30, 26 October 2009

2009-10-26T19:30:35Z

@@ Line 13: / Line 13: @@
 ===Stronger properties===
-* [[Weaker than::Normal space]]
+{| class="wikitable" border="1"
-* [[Weaker than::Completely regular space]]
+! property !! quick description !! proof of implication !! proof of strictness (reverse implication failure) !! intermediate notions
 ===Weaker properties===
-* [[Stronger than::Hausdorff space]]
+{| class="wikitable" border="1"
 ==Facts==
 Any connected Urysohn space with at least two points is uncountable (more precisely, its cardinality must be at least that of the continuum). This follows from the fact that its image under any continuous function must be connected, and hence the Urysohn function separating two points must be surjective to <math>[0,1]</math>. {{proofat|[[Connected Urysohn implies uncountable]]}}

Vipul at 16:46, 17 January 2009

2009-01-17T16:46:48Z

← Older revision		Revision as of 16:46, 17 January 2009
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	===Stronger properties===		===Stronger properties===

	* [[Normal space]]		* [[Weaker than::Normal space]]
	* [[Completely regular space]]		* [[Weaker than::Completely regular space]]

	===Weaker properties===		===Weaker properties===

	* [[Hausdorff space]]		* [[Stronger than::Hausdorff space]]

	==Facts==		==Facts==

	Any connected Urysohn space with at least two points is uncountable (more precisely, its cardinality must be at least that of the continuum). This follows from the fact that its image under any continuous function must be connected, and hence the Urysohn function separating two points must be surjective to <math>[0,1]</math>. {{proofat\|[[Connected Urysohn implies uncountable]]}}		Any connected Urysohn space with at least two points is uncountable (more precisely, its cardinality must be at least that of the continuum). This follows from the fact that its image under any continuous function must be connected, and hence the Urysohn function separating two points must be surjective to <math>[0,1]</math>. {{proofat\|[[Connected Urysohn implies uncountable]]}}

Vipul: 4 revisions

2008-05-11T19:59:51Z

4 revisions

← Older revision	Revision as of 19:59, 11 May 2008
(No difference)

Vipul at 18:34, 18 December 2007

2007-12-18T18:34:50Z

← Older revision		Revision as of 18:34, 18 December 2007
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	{{variationof\|Hausdorffness}}		{{variationof\|Hausdorffness}}

	{{~~separation-based topospace property~~}}		{{T family\|T2.5}}

	==Definition==		==Definition==