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  "body": "\n## Summary\n\nA continuous action of a topological group $G$ on a space $X$ is proper when the map $(g, x) \\mapsto (g \\cdot x, x)$ is a [[proper map|15XEBGNGHC]].\nInformally, group elements that move a compact set back onto itself form a compact set.\nProper actions have Hausdorff orbit spaces, closed orbits and compact stabilizers.\nThey are the right setting for quotients such as homogeneous spaces $G/H$ and for the action of discrete groups on manifolds.\n\n## Explanations\n\n### L1 human\n\nThink of a group moving the points of a space around.\nThe action is proper when only a \"small\", compact, part of the group can bring a given bounded region back near itself.\nInteger translations of the real line are proper: to move $[0, 1]$ to a place where it meets itself, only the shifts by $-1$, $0$ and $1$ work.\nScaling the line by positive numbers is not proper, because every scaling keeps the point $0$ in place.\n",
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