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  "body": "\n## Statement\n\nEach [[complete metric space|08EWT7J0SC]] is a [[Baire space|0D5KR33H7X]].\n\n## Remarks\n\nExplicitly: if $X$ is a complete metric space and $(U_n)_{n \\in \\mathbb{N}}$ is a sequence of open dense subsets of $X$, then $\\bigcap_n U_n$ is dense in $X$.\nThe proof builds a nested sequence of closed balls with radii tending to $0$, and uses completeness to find a point in all of them.\nA second form of the theorem says that each locally compact Hausdorff space is a Baire space.\nA consequence: $\\mathbb{R}$ is not a countable union of nowhere dense sets, and thus $\\mathbb{R}$ is uncountable.\n\n## Explanations\n\n### L1 human\n\nA complete metric space cannot be covered by countably many thin sets.\nFor example, you cannot write the real line as a countable union of sets like $\\mathbb{Z}$ or the Cantor set.\n",
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