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  "body": "\n## Summary\n\nAn analytic set is a subset of a [[Polish space|584GXXY73M]] that is a continuous image of the [[Baire space|P2WS16HKD7]].\nEquivalently, it is the projection of a closed set or of a Borel set in a product.\nEvery [[Borel set|0WQXY32Q8D]] is analytic, but some analytic sets are not Borel.\n[[Suslin's theorem|9418TGREZX]] says that a set is Borel exactly when both it and its complement are analytic.\nAnalytic sets are Lebesgue measurable, and an uncountable analytic set contains a perfect set.\n\n## Explanations\n\n### L1 human\n\nThe projection of a set in the plane onto the $x$-axis is its shadow.\nThe shadow of a closed set is simple, but the shadow of a complicated Borel set in the plane need not be Borel.\nAnalytic sets are exactly such shadows.\nThey are still well behaved: they can be measured, and they cannot be uncountable without containing a copy of the Cantor set.\n",
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