Collectionwise Hausdorff space: Difference between revisions
No edit summary |
|||
| Line 7: | Line 7: | ||
===Symbol-free definition=== | ===Symbol-free definition=== | ||
A [[topological space]] is said to be '''collectionwise Hausdorff''' if 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. | 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. | ||
===Definition with symbols=== | ===Definition with symbols=== | ||
| Line 23: | Line 23: | ||
* [[Hausdorff space]]: {{proofofstrictimplicationat|[[collectionwise Hausdorff implies Hausdorff]]|[[Hausdorff not implies collectionwise Hausdorff]]}} | * [[Hausdorff space]]: {{proofofstrictimplicationat|[[collectionwise Hausdorff implies Hausdorff]]|[[Hausdorff not implies collectionwise Hausdorff]]}} | ||
==Metaproperties== | |||
Revision as of 19:51, 17 December 2007
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
Symbol-free definition
A topological space is said to be collectionwise Hausdorff if it is 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.
Definition with symbols
Fill this in later
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