Functionally Hausdorff space: Difference between revisions

Revision as of 19:30, 26 October 2009

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

This is a variation of Hausdorffness. View other variations of Hausdorffness

In the T family (properties of topological spaces related to separation axioms), this is called: 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 $[0, 1]$ that takes the value $0$ at one point and $1$ at the other.

Relation with other properties

Stronger properties

property	quick description	proof of implication	proof of strictness (reverse implication failure)	intermediate notions
Normal space	$T_{1}$ and any two disjoint closed subsets are separated by disjoint open subsets	normal implies Urysohn	Urysohn not implies normal	Completely regular space\|FULL LIST, MORE INFO
Completely regular space	$T_{1}$ and continuous function to $[0, 1]$ separating any point and disjoint closed subset	completely regular implies Urysohn	Urysohn not implies completely regular	\|FULL LIST, MORE INFO

Weaker properties

property	quick description	proof of implication	proof of strictness (reverse implication failure)	intermediate notions
Hausdorff space	distinct points separated by disjoint open subsets	Urysohn implies Hausdorff	Hausdorff not implies Urysohn	\|FULL LIST, MORE INFO
T1 space	points are closed	(via Hausdorff)	(via Hausdorff)	Hausdorff space\|FULL LIST, MORE INFO

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 $[0, 1]$ . For full proof, refer: Connected Urysohn implies uncountable

Metaproperties

Metaproperty name	Satisfied?	Proof	Section in this article
subspace-hereditary property of topological spaces	Yes	Urysohn is hereditary	#Hereditariness
refining-preserved property of topological spaces	Yes	Urysohn is refining-preserved	#Refining
product-closed property of topological spaces	Yes	Urysohn is product-closed	#Products
box product-closed property of topological spaces	Yes	Urysohn is box product-closed	#Box products

Hereditariness

This property of topological spaces is hereditary, or subspace-closed. In other words, any subspace (subset with the subspace topology) of a topological space with this property also has this property.
View other subspace-hereditary properties of topological spaces

Any subspace of a Urysohn space is Urysohn. For full proof, refer: Urysohn is hereditary

Further information: Hausdorffness is hereditary, T1 is hereditary, regularity is hereditary, complete regularity is hereditary, normality is not hereditary

Refining

This property of topological spaces is preserved under refining, viz, if a set with a given topology has the property, the same set with a finer topology also has the property
View all refining-preserved properties of topological spaces OR View all coarsening-preserved properties of topological spaces

If $X$ is a Urysohn space with topology $τ$ , and $τ^{'}$ is a finer topology than $τ$ , then $(X, τ^{'})$ is also a Urysohn space. For full proof, refer: Urysohn is refining-preserved

Products

This property of topological spaces is closed under taking arbitrary products
View all properties of topological spaces closed under products

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. For full proof, refer: Urysohn is product-closed

Further information: Hausdorffness is product-closed

Box products

This property of topological spaces is a box product-closed property of topological spaces: it is closed under taking arbitrary box products
View other box product-closed properties of topological spaces

An arbitrary box product of Urysohn spaces, equipped wit the box topology, is also a Urysohn space. For full proof, refer: Urysohn is box product-closed

@@ 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 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 properties===
-* [[Stronger than::Hausdorff space]]
+{| class="wikitable" border="1"
+! property !! quick description !! 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::T1 space]] || points are closed || (via Hausdorff) || (via Hausdorff) || {{intermediate notions short|T1 space|Urysohn space}}
+|}
 ==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]]}}
+==Metaproperties==
+{| class="wikitable" border="1"
+!Metaproperty name !! Satisfied? !! Proof !! Section in this article
+|-
+|[[satisfies metaproperty::subspace-hereditary property of topological spaces]] || Yes || [[Urysohn is hereditary]] || [[#Hereditariness]]
+|-
+|[[satisfies metaproperty::refining-preserved property of topological spaces]] || Yes || [[Urysohn is refining-preserved]] || [[#Refining]]
+|-
+|[[satisfies metaproperty::product-closed property of topological spaces]]|| Yes || [[Urysohn is product-closed]] || [[#Products]]
+|-
+|[[satisfies metaproperty::box product-closed property of topological spaces]] || Yes || [[Urysohn is box product-closed]] || [[#Box products]]
+|}
+{{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]]}}