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  "body": "\n## Statement\n\nLet $X$ be a [[topological space|0CNQXVBHCX]].\nLet $(C_n)_{n \\ge 1}$ be a sequence of nonempty [[compact|00R0X1WGPX]] [[closed|0G5AV9Q4RR]] subsets of $X$ with $C_1 \\supseteq C_2 \\supseteq C_3 \\supseteq \\cdots$.\nThen the intersection $\\bigcap_{n \\ge 1} C_n$ is not empty.\n\n## Remarks\n\nThe sets must be compact: in $\\mathbb{R}$, the closed sets $[n, \\infty)$ are nested and nonempty, but their intersection is empty.\nFor $X = \\mathbb{R}$, the sets can be nested closed bounded intervals $[a_n, b_n]$.\nThere is a variant for metric spaces. A metric space is [[complete|08EWT7J0SC]] if and only if each nested sequence of nonempty closed sets whose [[diameters|0692CGQF00]] tend to $0$ has nonempty intersection. The intersection is then one point.\n\n## Explanations\n\n### L1 human\n\nIf you have a chain of nonempty compact sets, each inside the one before, at least one point is in all of them.\nFor example, the intervals $[0, 1/n]$ are nested, and $0$ is in all of them.\n",
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  "msc": [
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  "name": "Cantor's intersection theorem",
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