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  "body": "\n## Definition\n\nA [[topological space|0CNQXVBHCX]] $X$ is locally compact when each point $x \\in X$ has a [[neighbourhood|09CJ00H3NW]] $N$ that is [[compact|00R0X1WGPX]].\n\n## Remarks\n\nAuthors use several stronger definitions, for example that each neighbourhood of $x$ contains a compact neighbourhood of $x$. For [[Hausdorff spaces|0MXB4CKHFW]], all the usual definitions agree.\nEach compact space is locally compact. The space $\\mathbb{R}^n$ is locally compact, because each closed ball is compact.\nThe rationals $\\mathbb{Q}$ and each infinite-dimensional normed space are not locally compact.\nA locally compact Hausdorff space has a Hausdorff [[one-point compactification|0MM45BT6HC]].\n\n## Explanations\n\n### L1 human\n\nA locally compact space is compact in small pieces: each point has a compact set around it.\nThe real line is not compact, but each closed interval in it is.\n",
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