Baire space
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.