Functionally Hausdorff space: Difference between revisions

Latest revision as of 23:14, 27 January 2012

Definition

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 $[0, 1]$ that takes the value $0$ at one point and $1$ 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 $0$ at one point and $1$ 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.

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

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
normal Hausdorff space	$T_{1}$ and any two disjoint closed subsets are separated by disjoint open subsets	normal Hausdorff implies functionally Hausdorff	functionally Hausdorff not implies normal	\|FULL LIST, MORE INFO
Tychonoff space	$T_{1}$ and completely regular: continuous function to $[0, 1]$ separating any point and disjoint closed subset	Tychonoff implies functionally Hausdorff	functionally Hausdorff not implies completely regular	\|FULL LIST, MORE INFO
metrizable space				\|FULL LIST, MORE INFO
submetrizable space	either metrizable or can become metrizable upon passing to a coarser topology			\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Urysohn space	any two distinct points can be separated by disjoint open subsets whose closures are also disjoint			\|FULL LIST, MORE INFO
Hausdorff space	distinct points separated by disjoint open subsets			Urysohn space\|FULL LIST, MORE INFO
T1 space	points are closed	(via Hausdorff)	(via Hausdorff)	Hausdorff space\|FULL LIST, MORE INFO
Kolmogorov space	points are distinguishable	(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
subspace-hereditary property of topological spaces	Yes	functional Hausdorffness is hereditary
refining-preserved property of topological spaces	Yes	functional Hausdorffness is refining-preserved
product-closed property of topological spaces	Yes	functional Hausdorffness is product-closed
box product-closed property of topological spaces	Yes	functional Hausdorffness is box product-closed

@@ Line 1: / Line 1: @@
-{{topospace property}}
+==Definition==
-{{variationof|Hausdorffness}}
+A [[topological space]] is termed a '''completely Hausdorff space''' or '''functionally Hausdorff space''' if it satisfies the following equivalent conditions:
-{{T family|T2.5}}
+# 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.
-==Definition==
+{{topospace property}}
-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.
+{{variationof|Hausdorffness}}
 ==Relation with other properties==
@@ Line 13: / Line 16: @@
 ===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 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::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::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::metrizable space]] || || || || {{intermediate notions short|functionally Hausdorff space|metrizable 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::submetrizable space]] || either metrizable or can become metrizable upon passing to a coarser topology || || || {{intermediate notions short|functionally Hausdorff space|submetrizable space}}
 |}
 ===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::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 || [[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 || || || {{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::Kolmogorov space]] || points are distinguishable || (via Hausdorff) || (via Hausdorff) || {{intermediate notions short|Kolmogorov space|Urysohn space}}
 |}
@@ Line 38: / Line 49: @@
 {| 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 || [[functional Hausdorffness 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 || [[functional Hausdorffness 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 || [[functional Hausdorffness 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 || [[functional Hausdorffness 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]]}}