Weakly Hausdorff space: Difference between revisions
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| [[Weaker than::Hausdorff space]] || any two distinct points are separated by disjoint open subsets || || || {{intermediate notions short|weakly Hausdorff space|Hausdorff space}} | | [[Weaker than::Hausdorff space]] || any two distinct points are separated by disjoint open subsets || || || {{intermediate notions short|weakly Hausdorff space|Hausdorff space}} | ||
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| [[Stronger than::KC-space]] || every compact subset of the space is closed || || || {{intermediate notions short|KC-space|weakly Hausdorff 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 | ||
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| [[Stronger than::US-space]] || every sequence of points in the space has at most one limit || || || {{intermediate notions short|US-space|weakly Hausdorff space}} | | [[Stronger than::US-space]] || every sequence of points in the space has at most one limit || || || {{intermediate notions short|US-space|weakly Hausdorff space}} | ||
Latest revision as of 23:25, 17 July 2013
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
Definition
A topological space is termed a weak Hausdorff space' or weakly Hausdorff space if for every compact Hausdorff space and every continouus map , the image is a closed subset of .
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Hausdorff space | any two distinct points are separated by disjoint open subsets | |FULL LIST, MORE INFO | ||
| KC-space | every compact subset of the space is closed | |FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| US-space | every sequence of points in the space has at most one limit | KC-space|FULL LIST, MORE INFO | ||
| T1 space | every singleton subset is closed | KC-space|FULL LIST, MORE INFO |