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  "body": "\n## Summary\n\nAn action of a group by homeomorphisms is properly discontinuous when each compact set meets only finitely many of its translates.\nOn a locally compact Hausdorff space, it is the same as a proper action of the group with the discrete topology.\nA free properly discontinuous action of a group on a manifold gives a quotient manifold, and the quotient map is a [[covering map|3XZRRE165N]].\nExamples include lattices of translations acting on $\\R^n$, Fuchsian groups acting on the hyperbolic plane, and deck groups acting on universal covers.\n\n## Explanations\n\n### L1 human\n\nA group acts properly discontinuously if, from the point of view of any bounded region, only finitely many group elements move it to a nearby place.\nInteger translations of the plane tile it by unit squares, and a bounded piece of the plane meets only finitely many of its shifted copies.\nGluing points that differ by a group element then gives a nice space, here a torus.\nAn irrational rotation of the circle is the opposite case: every arc meets infinitely many of its rotated copies, and the quotient is a bad space.\n",
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