Functionally Hausdorff space: Difference between revisions

From Topospaces
No edit summary
No edit summary
Line 1: Line 1:
==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}}


Line 4: Line 13:


{{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}}
|}
|}


Line 38: Line 44:


{| 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]]}}

Revision as of 22:35, 24 January 2012

Definition

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

  1. 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.
  2. 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.
  3. 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.
  4. 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

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

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
normal space T1 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
nompletely regular space T1 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 Meaning 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
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 Urysohn is hereditary
refining-preserved property of topological spaces Yes Urysohn is refining-preserved
product-closed property of topological spaces Yes Urysohn is product-closed
box product-closed property of topological spaces Yes Urysohn is box product-closed
Retrieved from "https://topospaces.subwiki.org/w/index.php?title=Functionally_Hausdorff_space&oldid=4056"