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 $𝛼 : 𝐺 \times 𝑋 \to 𝑋$ if for all $𝑥, 𝑦 \in 𝑋$ there exists a $𝑔 \in 𝐺$ such that $𝛼 (𝑔, 𝑥) = 𝑦$ . #m/def/topology

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