Compactly homogeneous space

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.

Facts

• If a topological space is connected, Hausdorff and if every point has a compactly homogeneous neighbourhood, then the topological space is compactly homogeneous. Thus, the property of being compactly homogeneous is locally determined, unlike the property of being homogeneous.
• Euclidean space is compactly homogeneous, and hence, any connected manifold is compactly homogeneous.