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  "body": "\n## Statement\n\nThe [[Cantor set|0BXNDZFQME]] $C = \\bigcap_{n \\ge 0} C_n$ is a [[closed subset|0G5AV9Q4RR]] of $\\mathbb{R}$.\n\n## Remarks\n\nProof: each $C_n$ is a union of $2^n$ closed intervals, so it is closed. An [[intersection of closed sets is closed|09YMV8H4X5]].\nSince $C \\subseteq [0, 1]$, the set $C$ is also bounded, and thus compact by the [[Heine–Borel theorem|07PFYV65MB]].\nEach endpoint of an interval of some $C_n$ lies in $C$. For example, $\\tfrac{1}{3}$ and $\\tfrac{2}{3}$ are in $C$.\n\n## Explanations\n\n### L1 human\n\nAt each step we keep a finite union of closed intervals, which is a closed set.\nClosed sets stay closed under intersections, so the limit set is closed.\n",
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