[[Topology MOC]]
# Homogenous space

A **homogenous space** $X$ is a [[topological space]] such that for any $x,y \in X$ there exists an automorphism $f : X \to X$ such thar $f(x)=y$. #m/def/topology 
Thus the space ‘looks the same’ everywhere.

## Homogeneity under an action

A topological space $X$ is homogeneous under a [[group action]] $\alpha : G \times X \to X$ if for all $x,y \in X$ there exists a $g \in G$ such that $\alpha(g,x) = y$. #m/def/topology 

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