---
tag: 3RWWZYPAFV
kind: concept
name: Analytic set
msc: ["03E15", "28A05", "54H05"]
sources:
  - url: https://en.wikipedia.org/wiki/Analytic_set
same_as:
  wikidata: Q485312
  wikipedia: Analytic set
---

## Summary

An analytic set is a subset of a [[Polish space|584GXXY73M]] that is a continuous image of the [[Baire space|P2WS16HKD7]].
Equivalently, it is the projection of a closed set or of a Borel set in a product.
Every [[Borel set|0WQXY32Q8D]] is analytic, but some analytic sets are not Borel.
[[Suslin's theorem|9418TGREZX]] says that a set is Borel exactly when both it and its complement are analytic.
Analytic sets are Lebesgue measurable, and an uncountable analytic set contains a perfect set.

## Explanations

### L1 human

The projection of a set in the plane onto the $x$-axis is its shadow.
The shadow of a closed set is simple, but the shadow of a complicated Borel set in the plane need not be Borel.
Analytic sets are exactly such shadows.
They are still well behaved: they can be measured, and they cannot be uncountable without containing a copy of the Cantor set.