In the mathematical field of descriptive set theory, a subset A of a Polish space X is projective if it is Σ n 1 for some positive integer n . Here A is
Σ 1 1 if A is analytic Π n 1 if the complement of A , X ∖ A , is Σ n 1 Σ n + 1 1 if there is a Polish space Y and a Π n 1 subset C ⊆ X × Y such that A is the projection of C ; that is, A = { x ∈ X ∣ ∃ y ∈ Y ( x , y ) ∈ C } The choice of the Polish space Y in the third clause above is not very important; it could be replaced in the definition by a fixed uncountable Polish space, say Baire space or Cantor space or the real line.
There is a close relationship between the relativized analytical hierarchy on subsets of Baire space (denoted by lightface letters Σ and Π ) and the projective hierarchy on subsets of Baire space (denoted by boldface letters Σ and Π ). Not every Σ n 1 subset of Baire space is Σ n 1 . It is true, however, that if a subset X of Baire space is Σ n 1 then there is a set of natural numbers A such that X is Σ n 1 , A . A similar statement holds for Π n 1 sets. Thus the sets classified by the projective hierarchy are exactly the sets classified by the relativized version of the analytical hierarchy. This relationship is important in effective descriptive set theory.
A similar relationship between the projective hierarchy and the relativized analytical hierarchy holds for subsets of Cantor space and, more generally, subsets of any effective Polish space.
