Compactly homogeneous space: Difference between revisions
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A [[topological space]] is termed '''compactly homogeneous''' if it is [[connected space|connected]], and given any two points, there is an open set containing them, whose closure is compact, and such that there is a homeomorphism of the topological space which sends one point to the other, and is identity outside the open set. | A [[topological space]] is termed '''compactly homogeneous''' if it is [[connected space|connected]], and given any two points, there is an open set containing them, whose closure is compact, and such that there is a homeomorphism of the topological space which sends one point to the other, and is identity outside the open set. | ||
==Relation with other properties== | |||
===Stronger properties=== | |||
* [[Euclidean space]] | |||
===Weaker properties=== | |||
* [[Homogeneous space]] | |||
==Facts== | ==Facts== | ||
Revision as of 18:30, 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
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Definition
Symbol-free definition
A topological space is termed compactly homogeneous if it is connected, and given any two points, there is an open set containing them, whose closure is compact, and such that there is a homeomorphism of the topological space which sends one point to the other, and is identity outside the open set.
Relation with other properties
Stronger properties
Weaker properties
Facts
- If a topological space is connected, Hausdorff and if every point has a compactly homogeneous neighbourhood, then the topological space is homogeneous.
- Euclidean space is compactly homogeneous, and hence, any connected manifold is compactly homogeneous.