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  "body": "\n## Proof\n\n1. Let $X$ be a [[topological space|0CNQXVBHCX]] and let $U \\subseteq X$.\n2. By the definition of a [[closed set|0G5AV9Q4RR]], the set $X \\setminus U$ is closed if and only if its [[complement|0B1FSS1VZW]] $X \\setminus (X \\setminus U)$ is [[open|01ZQQDCERP]].\n3. A point $x \\in X$ lies in $X \\setminus (X \\setminus U)$ exactly when it is not outside $U$, that is, when $x \\in U$. So $X \\setminus (X \\setminus U) = U$.\n4. By steps 2 and 3, $X \\setminus U$ is closed if and only if $U$ is open.\n\n## Idea\n\nA set is closed when its complement is open. Taking the complement twice gives back the set, so the statement is the definition read the other way round.\n",
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