Functionally Hausdorff space: Difference between revisions
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==Definition== | ==Definition== | ||
A [[topological space]] is termed a '''completely Hausdorff space''' or ''' | 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}} | ||
==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 | | [[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:: | | [[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| | | [[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| | | [[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:: | | [[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}} | ||
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!Metaproperty name !! Satisfied? !! Proof | !Metaproperty name !! Satisfied? !! Proof | ||
|- | |- | ||
|[[satisfies metaproperty::subspace-hereditary property of topological spaces]] || Yes || [[ | |[[satisfies metaproperty::subspace-hereditary property of topological spaces]] || Yes || [[functional Hausdorffness is hereditary]] | ||
|- | |- | ||
|[[satisfies metaproperty::refining-preserved property of topological spaces]] || Yes || [[ | |[[satisfies metaproperty::refining-preserved property of topological spaces]] || Yes || [[functional Hausdorffness is refining-preserved]] | ||
|- | |- | ||
|[[satisfies metaproperty::product-closed property of topological spaces]]|| Yes || [[ | |[[satisfies metaproperty::product-closed property of topological spaces]]|| Yes || [[functional Hausdorffness is product-closed]] | ||
|- | |- | ||
|[[satisfies metaproperty::box product-closed property of topological spaces]] || Yes || [[ | |[[satisfies metaproperty::box product-closed property of topological spaces]] || Yes || [[functional Hausdorffness is box product-closed]] | ||
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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 that takes the value at one point and 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 at one point and 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 | 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 | and completely regular: continuous function to 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 . For full proof, refer: Connected Urysohn implies uncountable