Analytic Set

In the mathematical field of descriptive set theory, a subset of a Polish space $X$ 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 ω^ω.
*''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.
*There is a Polish space $Y$ and a Borel set $B\subseteq X\times Y$ such that $A$ is the projection of $B$ onto $X$; that is,
: $A=\.$
*''A'' is the projection of a closed set in the cartesian product of ''X'' with the Baire space.
*''A'' is the projection of a G_δ set in the cartesian product of ''X'' with the Cantor space 2^ω.

An alternative characterization, in the specific, important, case that $X$ is Baire space ω^ω, is that the analytic sets are precisely the projections of trees on $\omega\times\omega$. Similarly, the analytic subsets of Cantor space 2^ω are precisely the projections of trees on $2\times\omega$.

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