Relationship To The Analytical Hierarchy
There is a close relationship between the relativized analytical hierarchy on subsets of Baire space and the projective hierarchy on subsets of Baire space. Not every subset of Baire space is . It is true, however, that if a subset X of Baire space is then there is a set of natural numbers A such that X is . A similar statement holds for 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.
Read more about this topic: Projective Hierarchy
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