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 Luzin (1917) and his student Souslin (1917).
[1] 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
Analytic subsets of Polish spaces are closed under countable unions and intersections, continuous images, and inverse images.
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.
Note that the bold font in this symbol is not the Wikipedia convention, but rather is used distinctively from its lightface counterpart