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  "body": "\n## Statement\n\nLet $(X, d)$ be a [[metric space|063W5JMY85]] and let $(F_i)_{i \\in I}$ be a family of [[closed subsets|0G5AV9Q4RR]] of $X$, indexed by a nonempty set $I$.\nThen the [[intersection|0839XNQXDK]] $\\bigcap_{i \\in I} F_i$ is closed in $X$.\n\n## Remarks\n\nProof: by [[De Morgan's laws for sets|0VK0W4D5Y5]], $X \\setminus \\bigcap_i F_i = \\bigcup_i (X \\setminus F_i)$. This is a [[union of open sets|0HEF46RPEK]], so it is open.\nThe same proof works in each [[topological space|0CNQXVBHCX]].\nA union of finitely many closed sets is closed, but an infinite union need not be. For example, the union of the sets $[1/n, 1]$ for $n \\ge 1$ is $(0, 1]$.\n\n## Explanations\n\n### L1 human\n\nClosed sets behave like open sets with the roles of union and intersection exchanged.\nAny intersection of closed sets is closed, and finite unions of closed sets are closed.\n",
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