Locally Hausdorff space: Difference between revisions
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===Stronger properties=== | ===Stronger properties=== | ||
* [[Hausdorff space]] | * [[Hausdorff space]]: {{proofofstrictimplicationat|[[Hausdorff implies locally Hausdorff]]|[[locally Hausdorff not implies Hausdorff]]}} | ||
* [[Locally metrizable space]] | * [[Locally metrizable space]] | ||
* [[Locally Euclidean space]] | * [[Locally Euclidean space]] | ||
Revision as of 22:44, 15 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 termed locally Hausdorff if it satisfies the following equivalent conditions:
- Every point has an open neighbourhood which is Hausdorff
- Given any point, and any open neighbourhood of the point, there is a smaller open neighbourhood of the point which is Hausdorff.
Relation with other properties
Stronger properties
- Hausdorff space: For proof of the implication, refer Hausdorff implies locally Hausdorff and for proof of its strictness (i.e. the reverse implication being false) refer locally Hausdorff not implies Hausdorff
- Locally metrizable space
- Locally Euclidean space