Topological homogeneity

Remarks

Example: every topological group $G$ is homogeneous. For $x, y \in G$, left translation by $y x^{-1}$ is a homeomorphism with inverse the translation by $x y^{-1}$, and it sends $x$ to $y$. Non-example: the closed interval $[0, 1]$. A homeomorphism $h$ with $h(0) = 1/2$ would restrict to a homeomorphism from $(0, 1]$, which is connected, onto $[0, 1] \setminus \set{1/2}$, which is not connected. In the language of group actions, $X$ is homogeneous when its homeomorphism group acts transitively on $X$.