Analytic set

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This article is about analytic sets as defined in descriptive set theory. There is another notion in the context of analytic varieties.

Given a Polish space <math>X</math>, a subset <math>A\subseteq X</math> is analytic if there is a Polish space <math>Y</math> and a Borel set <math>B\subseteq X\times Y</math> such that <math>A</math> is the projection of <math>B</math>; that is,

<math>A=\{x\in X|(\exists y\in Y)<x,y>\in B\}.</math>

Note that the choice of the Polish space <math>Y</math> above is not very important; it could be replaced in the definition by a fixed uncountable Polish space, say Baire space or Cantor space or the real line.

An alternative characterization, in the specific (and important) case that <math>X</math> is Baire space, is that the analytic sets are precisely the projections of trees on <math>\omega\times\omega</math>. Similarly, the analytic subsets of Cantor space are precisely the projections of trees on <math>2\times\omega</math>.

Analytic sets are also called <math>\boldsymbol{\Sigma}^1_1</math> (see projective hierarchy). Note that the bold font in this symbol is not the Wikipedia convention, but rather is used distinctively from its lightface counterpart <math>\Sigma^1_1</math> (see analytical hierarchy).

Analytic sets are always Lebesgue measurable (indeed, universally measurable) and have the property of Baire and the perfect set property.