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  "body": "\n## Definition\n\nA [[metric space|063W5JMY85]] $(X, d)$ is proper when each [[closed ball|05DSV40EB9]] $\\bar{B}(x, r)$, with $x \\in X$ and $r \\ge 0$, is [[compact|00R0X1WGPX]].\n\n## Remarks\n\nIn a proper metric space, a subset is compact if and only if it is closed and bounded. Thus the [[Heine–Borel theorem|07PFYV65MB]] says that $\\mathbb{R}^n$ is proper.\nEach proper metric space is complete and [[locally compact|0EQMWQ4WMJ]].\nAn infinite set with the [[discrete metric|0NA4FK7B8S]] is not proper: the closed ball of radius $1$ is the whole set, which is not compact.\n\n## Explanations\n\n### L1 human\n\nA proper metric space is a space where the Heine–Borel test for compactness works: closed and bounded is enough.\nEuclidean spaces are proper, but many infinite-dimensional spaces are not.\n",
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