Analytic set

Properties

Analytic subsets of Polish spaces are closed under countable unions and intersections, continuous images, and inverse images. The complement of an analytic set need not be analytic. Suslin proved that if the complement of an analytic set is analytic then the set is Borel. (Conversely any Borel set is analytic and Borel sets are closed under complements.) Luzin proved more generally that any two disjoint analytic sets are separated by a Borel set: in other words there is a Borel set including one and disjoint from the other. This is sometimes called the "Luzin separability principle" (though it was implicit in the proof of Suslin's theorem).

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

Examples

When ${\displaystyle A}$ is a set of natural numbers, refer to the set ${\displaystyle \{x-y\mid y\le x\wedge x,y\in A\}}$ as the difference set of ${\displaystyle A}$. The set of difference sets of natural numbers is an analytic set, and is complete for analytic sets.

Projective hierarchy

Analytic sets are also called ${\displaystyle {\mathbf{\Sigma }}_1^1}$ (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 ${\displaystyle {\mathrm{\Sigma }}_1^1}$ (see analytical hierarchy). The complements of analytic sets are called coanalytic sets, and the set of coanalytic sets is denoted by ${\displaystyle {\mathbf{\Pi }}_1^1}$. The intersection ${\displaystyle {\mathbf{\Delta }}_1^1={\mathbf{\Sigma }}_1^1\cap {\mathbf{\Pi }}_1^1}$ is the set of Borel sets.