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Concept
Projective hierarchy
Summary
In the mathematical field of descriptive set theory, a subset of a Polish space is projective if it is for some positive integer . Here is
if is analytic
if the complement of , , is
if there is a Polish space and a subset such that is the projection of onto ; that is,
The choice of the Polish space 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 and ) and the projective hierarchy on subsets of Baire space (denoted by boldface letters and ). 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.
Official source
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