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  "body": "\n## Statement\n\nLet $X$ and $Y$ be [[topological spaces|0CNQXVBHCX]], and let $K \\subseteq X$ and $L \\subseteq Y$ be [[compact|00R0X1WGPX]].\nThen $K \\times L$ is compact in $X \\times Y$ with the [[product topology|0Z57EC6YX3]].\n\n## Remarks\n\nThe proof uses the tube lemma. For each $x \\in K$, finitely many basic open sets $U \\times V$ cover $\\{x\\} \\times L$, and their intersection over the first factor gives an open set about $x$.\nBy induction, a product of finitely many compact sets is compact. For example, $[0, 1]^n$ is compact.\nFor arbitrary products of compact spaces, the statement is [[Tychonoff's theorem|0JTRSJKCFB]].\n\n## Explanations\n\n### L1 human\n\nA rectangle made of two compact sides is compact.\nFor example, the square $[0, 1] \\times [0, 1]$ is compact, because $[0, 1]$ is compact.\n",
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