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  "body": "\n## Summary\n\nThe Borel hierarchy sorts the Borel sets of a [[Polish space|584GXXY73M]] into levels $\\Sigma^0_\\alpha$, $\\Pi^0_\\alpha$ and $\\Delta^0_\\alpha$, indexed by the countable [[ordinals|0YSCSX5YZC]] $\\alpha \\ge 1$.\nIt starts with the open and the closed sets, and each new level allows countable unions of complements of earlier sets.\nEvery [[Borel set|0WQXY32Q8D]] appears at some countable level, and in an uncountable Polish space every level contains new sets.\nThe level of a set measures how complicated it is, and this is a basic tool of [[descriptive set theory|TF8HMVXKYC]].\n\n## Explanations\n\n### L1 human\n\nOpen and closed sets are the simplest sets.\nCountable unions of closed sets, and countable intersections of open sets, are one step more complex.\nRepeating this, one climbs a ladder of complexity with one rung for each countable ordinal.\nFor example, the set of rational numbers is a countable union of points, so it sits on the second rung, but it is not a countable intersection of open sets.\n",
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