Projective hierarchy

In the mathematical field of descriptive set theory, a subset formula_1 of a Polish space formula_2 is projective if it is formula_3 for some positive integer formula_4. Here formula_1 is
The choice of the Polish space formula_13 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.
Relationship to the analytical hierarchy.
There is a close relationship between the relativized analytical hierarchy on subsets of Baire space (denoted by lightface letters formula_20 and formula_21) and the projective hierarchy on subsets of Baire space (denoted by boldface letters formula_22 and formula_23). Not every formula_3 subset of Baire space is formula_25. It is true, however, that if a subset "X" of Baire space is formula_3 then there is a set of natural numbers "A" such that "X" is formula_27. A similar statement holds for formula_8 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.