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  "body": "\n## Definition\n\nLet $\\Gamma$ be a group and let $X$ be a topological space.\nLet $\\Gamma$ [[act|0WCY42EV9V]] on $X$ by [[homeomorphisms|0QT00PB5G2]].\nThe action is *properly discontinuous* if, for all [[compact|0TVJZYB6SH]] subsets $K, L \\subseteq X$, the set\n$$\\set{\\gamma \\in \\Gamma : (\\gamma \\cdot K) \\cap L \\ne \\emptyset}$$\nis finite.\n\n## Remarks\n\nWhen $X$ is Hausdorff and locally compact, the action is properly discontinuous if and only if the action of $\\Gamma$ with the [[discrete topology|0S2TEHX4E1]] is [[proper|MDH7VXDCVF]].\nIn covering space theory, some authors use the term for the weaker condition that each point has a neighbourhood $U$ with $\\gamma U \\cap U = \\emptyset$ for all $\\gamma \\ne 1$.\n\nExample: let $\\Z^n$ act on $\\R^n$ by translations.\nIf $K$ and $L$ lie in the ball of radius $R$ about $0$, then $(v + K) \\cap L \\ne \\emptyset$ forces $\\norm{v} \\le 2R$.\nOnly finitely many $v \\in \\Z^n$ satisfy this, so the action is properly discontinuous.\n\nNon-example: let $\\Z$ act on $\\R$ by $n \\cdot x = 2^n x$.\nFor $K = L = \\set{0}$, every $n \\in \\Z$ satisfies $(n \\cdot K) \\cap L \\ne \\emptyset$, so the set is infinite.\nSo the action is not properly discontinuous.\n",
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