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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]].\nA subset $A \\subseteq X$ is a *$G_\\delta$ set* if there are [[open sets|02BH3MCBQB]] $U_1, U_2, U_3, \\dots$ of $X$ with\n$$A = \\bigcap_{n=1}^{\\infty} U_n.$$\n\n## Remarks\n\nEvery open set is $G_\\delta$: take $U_n = A$ for all $n$.\nIn a [[metric space|0VKD1TP7F3]] with metric $d$, every [[closed set|0ZCK8DQYN1]] $F$ is $G_\\delta$, since $F = \\bigcap_n \\set{x : d(x, F) \u003c 1/n}$.\n\nExample: the set of irrational numbers is $\\bigcap_{q \\in \\Q} (\\R \\setminus \\set{q})$.\nEach $\\R \\setminus \\set{q}$ is open and $\\Q$ is [[countable|01TQB88H4C]], so the irrationals form a $G_\\delta$ set.\n\nNon-example: $\\Q$ is not a $G_\\delta$ subset of $\\R$.\nIf $\\Q = \\bigcap_n U_n$ with $U_n$ open, each $U_n$ is open and dense, and so are the sets $\\R \\setminus \\set{q}$ for $q \\in \\Q$.\nThese countably many dense open sets have empty intersection, which contradicts the [[Baire category theorem|0A7KGAYHGV]].\n\nA set is $G_\\delta$ if and only if its complement is an [[F-sigma set|8MGC7M64GW]].\n",
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