Analytic set


In the mathematical field of descriptive set theory, a subset of a Polish space is an analytic set if it is a continuous image of a Polish space. These sets were first defined by and his student.

Definition

There are several equivalent definitions of analytic set. The following conditions on a subspace A of a Polish space X are equivalent:
An alternative characterization, in the specific, important, case that is Baire space ωω, is that the analytic sets are precisely the projections of trees on. Similarly, the analytic subsets of Cantor space 2ω are precisely the projections of trees on.

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. Luzin proved more generally that any two disjoint analytic sets are separated by a Borel set: in other words there is a Borel set containing one and disjoint from the other. This is sometimes called the "Luzin separability principle".
Analytic sets are always Lebesgue measurable and have the property of Baire and the perfect set property.

Projective hierarchy

Analytic sets are also called . Note that the bold font in this symbol is not the Wikipedia convention, but rather is used distinctively from its lightface counterpart . The complements of analytic sets are called coanalytic sets, and the set of coanalytic sets is denoted by.
The intersection is the set of Borel sets.