- Projective hierarchy
In the mathematical field of
descriptive set theory , a subset A of aPolish space X is projective if it is oldsymbol{Sigma}^1_n for some positive integer n. Here A is
* oldsymbol{Sigma}^1_1 if A is analytic
* oldsymbol{Pi}^1_n if thecomplement of A, Xsetminus A, is oldsymbol{Sigma}^1_n
* oldsymbol{Sigma}^1_{n+1} if there is a Polish space Y and a oldsymbol{Pi}^1_n subset Csubseteq X imes Y such that A is the projection of C; that is, A={xin X|(exists yin Y){langle}x,y{ angle}in 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 thereal line .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 oldsymbol{Sigma}^1_n subset of Baire space is Sigma^1_n. It is true, however, that if a subset "X" of Baire space is oldsymbol{Sigma}^1_n then there is a set of natural numbers "A" such that "X" is Sigma^{1,A}_n. A similar statement holds for oldsymbol{Pi}^1_n 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 ineffective 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 .References
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