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  "body": "\n## Summary\n\nA topological space is homogeneous when for any two of its points some [[homeomorphism|0QT00PB5G2]] of the space carries the first point to the second.\nSo the space looks the same near every point.\nEvery [[topological group|92700QAB5Y]] is homogeneous, because translations are homeomorphisms.\nHomogeneity is therefore a first test of whether a space can carry the structure of a topological group.\n\n## Explanations\n\n### L1 human\n\nA homogeneous space has no special points: whatever you can say about the space near one point, you can say near any other.\nThe circle is homogeneous, since rotating it moves any point to any other.\nA closed interval $[0, 1]$ is not homogeneous: an endpoint is different from an inner point.\nRemoving an endpoint leaves the interval connected, while removing an inner point cuts it in two.\nSo a closed interval cannot be made into a topological group.\n",
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