# Compactly homogeneous space

From Topospaces

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 term is nonstandard and is being used locally within the wiki. For its use outside the wiki, please define the term when using it.*

## Contents

## 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

- Euclidean space:
*For full proof, refer: Euclidean implies compactly homogeneous*

### 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.