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:

• A is analytic.

• A is empty or a continuous image of the Baire space ωω.

• A is a Suslin space, in other words A is the image of a Polish space under a continuous mapping.

• A is the continuous image of a Borel set in a Polish space.

• A is a Suslin set, the image of the Suslin operation.

• and a Borel is the projection; that is,

• A is the projection of a closed set in the cartesian product of X with the Baire space.

• A is the projection of a Gδ set in the cartesian product of X with the Cantor space 2ω.

is Baire space ωω, is that the analytic sets are precisely the projections of trees. Similarly, the analytic subsets of Cantor space 2ω.

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

. The set of difference sets of natural numbers is an analytic set, and is complete for analytic sets.

Projective hierarchy

(see projective hierarchy (see analytical hierarchy). The complements of analytic sets are called coanalytic sets is the set of Borel sets.

See also

• Projection (measure theory)

References

• N.N. Lusin, "Leçons sur les ensembles analytiques et leurs applications", Gauthier-Villars (1930)

• Martin, Donald A.: Measurable cardinals and analytic games. Fundamenta Mathematicae 66 (1969/1970), p.287-291.