Collectionwise Hausdorff space: Difference between revisions
No edit summary |
|||
| (3 intermediate revisions by the same user not shown) | |||
| Line 5: | Line 5: | ||
==Definition== | ==Definition== | ||
A [[topological space]] is said to be '''collectionwise Hausdorff''' if it satisfies the following: it is [[T1 space|T1]] and given any discrete [[closed subset]] (viz a closed subset that is a [[discrete space]] under the induced topology), we can find a disjoint family of open sets, with each point of the discrete subset contained in exactly one member open set. | |||
A [[topological space]] is said to be '''collectionwise Hausdorff''' if it is [[T1 space|T1]] and given any discrete [[closed subset]] (viz a closed subset that is discrete under the induced topology), we can find a disjoint family of open sets, with each point of the discrete subset contained in exactly one member open set. | |||
==Relation with other properties== | ==Relation with other properties== | ||
| Line 17: | Line 11: | ||
===Stronger properties=== | ===Stronger properties=== | ||
* [[Metrizable | * [[Weaker than::Metrizable space]] | ||
===Weaker properties=== | ===Weaker properties=== | ||
* [[Hausdorff space]]: {{proofofstrictimplicationat|[[collectionwise Hausdorff implies Hausdorff]]|[[Hausdorff not implies collectionwise Hausdorff]]}} | * [[Stronger than::Hausdorff space]]: {{proofofstrictimplicationat|[[collectionwise Hausdorff implies Hausdorff]]|[[Hausdorff not implies collectionwise Hausdorff]]}} | ||
==Metaproperties== | ==Metaproperties== | ||
Latest revision as of 23:47, 15 November 2015
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
Definition
A topological space is said to be collectionwise Hausdorff if it satisfies the following: it is T1 and given any discrete closed subset (viz a closed subset that is a discrete space under the induced topology), we can find a disjoint family of open sets, with each point of the discrete subset contained in exactly one member open set.
Relation with other properties
Stronger properties
Weaker properties
- Hausdorff space: For proof of the implication, refer collectionwise Hausdorff implies Hausdorff and for proof of its strictness (i.e. the reverse implication being false) refer Hausdorff not implies collectionwise Hausdorff