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  "body": "\n## Summary\n\nA $G_\\delta$ set in a [[topological space|0CQHFPGJ45]] is an intersection of countably many [[open sets|02BH3MCBQB]].\nThe name comes from the German words for open set and intersection.\nIn a [[metric space|0VKD1TP7F3]] every closed set is $G_\\delta$, and $G_\\delta$ sets are the next step after open and closed sets in the Borel hierarchy.\nThe complements of $G_\\delta$ sets are the [[F-sigma sets|8MGC7M64GW]].\n\n## Explanations\n\n### L1 human\n\nOne open set may be large, but intersecting countably many open sets can carve out a much thinner set.\nThe irrational numbers form a $G_\\delta$ set: for each rational $q$, remove the single point $q$ from $\\R$, which leaves an open set, and intersect all these sets.\nOn the other hand, the rationals are not $G_\\delta$ in $\\R$, which follows from the [[Baire category theorem|0A7KGAYHGV]].\n",
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