Preregular space: Difference between revisions
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==Definition== | ==Definition== | ||
A [[topological space]] is termed '''preregular''' if | A [[topological space]] is termed '''preregular''' if it satisfies the following equivalent conditions: | ||
# Any two [[defining ingredient::topological indistinguishability|topologically distinguishable points]] can be separated by pairwise disjoint [[open subset]]s. | |||
# Its [[defining ingredient::Kolmogorov quotient]] is a [[defining ingredient::Hausdorff space]]. | |||
{{topospace property}} | {{topospace property}} | ||
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| [[Weaker than::regular space]] || || || || {{intermediate notions short|preregular space|regular space}} | | [[Weaker than::regular space]] || || || || {{intermediate notions short|preregular space|regular space}} | ||
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===Weaker properties=== | |||
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! Property !! Meaning!! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
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| [[Stronger than::symmetric space]] || [[Kolmogorov quotient]] is a [[T1 space]]. || || || {{intermediate notions short|symmetric space|preregular space}} | |||
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Latest revision as of 15:36, 28 January 2012
Definition
A topological space is termed preregular if it satisfies the following equivalent conditions:
- Any two topologically distinguishable points can be separated by pairwise disjoint open subsets.
- Its Kolmogorov quotient is a Hausdorff space.
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
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Hausdorff space | |FULL LIST, MORE INFO | |||
| regular space | |FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| symmetric space | Kolmogorov quotient is a T1 space. | |FULL LIST, MORE INFO |