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  "body": "\n## Definition\n\nA [[topological space|0CQHFPGJ45]] $X$ is *homogeneous* if for all $x, y \\in X$ there is a [[homeomorphism|0QT00PB5G2]] $h \\colon X \\to X$ with $h(x) = y$.\n\n## Remarks\n\nExample: every [[topological group|92700QAB5Y]] $G$ is homogeneous.\nFor $x, y \\in G$, left translation by $y x^{-1}$ is a homeomorphism with inverse the translation by $x y^{-1}$, and it sends $x$ to $y$.\nNon-example: the closed interval $[0, 1]$.\nA 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.\nIn the language of group actions, $X$ is homogeneous when its homeomorphism group acts transitively on $X$.\n",
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  "name": "Topological homogeneity",
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