An alternative characterization, in the specific, important, case that is Baire space ??, is that the analytic sets are precisely the projections of trees on . Similarly, the analytic subsets of Cantor space 2? are precisely the projections of trees on .
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 containing 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 also called (see projective hierarchy). Note that the bold font in this symbol is not the popflock.com resource convention, but rather is used distinctively from its lightface counterpart (see analytical hierarchy). 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.