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  "body": "\n## Definition\n\nLet $G$ be a [[topological group|92700QAB5Y]] and let $X$ be a topological space.\nLet $G$ act continuously on $X$, that is, let a [[group action|0WCY42EV9V]] $G \\times X \\to X$, $(g, x) \\mapsto g \\cdot x$, be continuous.\nThe action is *proper* if the map\n$$\\theta \\colon G \\times X \\to X \\times X, \\qquad (g, x) \\mapsto (g \\cdot x, x)$$\nis a [[proper map|15XEBGNGHC]], that is, continuous, closed, and with compact fibres.\n\n## Remarks\n\nWhen $G$ and $X$ are Hausdorff and [[locally compact|0FE2SB8WVJ]], $\\theta$ is proper if and only if the preimage of every [[compact|0TVJZYB6SH]] set is compact.\nThen the action is proper if and only if, for every compact $K \\subseteq X$, the set $\\set{g \\in G : (g \\cdot K) \\cap K \\ne \\emptyset}$ is compact.\n\nExample: let $G = \\Z$ act on $X = \\R$ by $n \\cdot x = x + n$.\nFor compact $K \\subseteq [-R, R]$, the condition $(n + K) \\cap K \\ne \\emptyset$ forces $\\abs{n} \\le 2R$, so the set of such $n$ is finite, hence compact in $\\Z$.\nSo the action is proper.\n\nNon-example: let $G = (0, \\infty)$ under multiplication act on $X = \\R$ by $t \\cdot x = tx$.\nFor $K = [-1, 1]$, the point $0$ lies in $t \\cdot K \\cap K$ for every $t$, so the set of such $t$ is all of $(0, \\infty)$, which is not compact.\nSo the action is not proper.\n\nEvery continuous action of a compact Hausdorff group on a Hausdorff space is proper.\n",
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