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  "body": "\n## Statement\n\nLet $X$ be a [[topological space|0CNQXVBHCX]] and let $U \\subseteq X$.\nThen $U$ is [[open|01ZQQDCERP]] in $X$ if and only if $X \\setminus U$ is [[closed|0G5AV9Q4RR]] in $X$.\n\n## Remarks\n\nProof: $X \\setminus (X \\setminus U) = U$, so the claim is the definition of a closed set applied to $X \\setminus U$.\nThus the closed sets determine the open sets, and the open sets determine the closed sets.\n\n## Explanations\n\n### L1 human\n\nTaking the complement turns open sets into closed sets and closed sets back into open sets.\nFor example, $(0, 1)$ is open in $\\mathbb{R}$, so $(-\\infty, 0] \\cup [1, \\infty)$ is closed.\n",
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