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In the mathematical field of [[descriptive set theory]], a subset <math>A</math> of a [[Polish space]] <math>X</math> is '''projective''' if it is <math>\boldsymbol{\Sigma}^1_n</math> for some positive integer <math>n</math>. Here <math>A</math> is
* <math>\boldsymbol{\Sigma}^1_1</math> if <math>A</math> is [[analytic set|analytic]]
* <math>\boldsymbol{\Pi}^1_n</math> if the [[Complement (set theory)|complement]] of <math>A</math>, <math>X\setminus A</math>, is <math>\boldsymbol{\Sigma}^1_n</math>
* <math>\boldsymbol{\Sigma}^1_{n+1}</math> if there is a Polish space <math>Y</math> and a <math>\boldsymbol{\Pi}^1_n</math> subset <math>C\subseteq X\times Y</math> such that <math>A</math> is the projection of <math>C</math>; that is, <math>A=\{x\in X|(\exists y\in Y){\langle}x,y{\rangle}\in C\}</math>


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The choice of the Polish space <math>Y</math> 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 (set theory)|Baire space]] or [[Cantor space]] or the [[real line]].
 
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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 <math>\boldsymbol{\Sigma}^1_n</math> subset of Baire space is <math>\Sigma^1_n</math>.   It is true, however, that if a subset ''X'' of Baire space is <math>\boldsymbol{\Sigma}^1_n</math> then there is a set of natural numbers ''A'' such that ''X'' is <math>\Sigma^{1,A}_n</math>.   A similar statement holds for <math>\boldsymbol{\Pi}^1_n</math> 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]].
 
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== References ==
 
* {{Citation | last1=Kechris | first1=A. S. | title=Classical Descriptive Set Theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-94374-9 | year=1995}}
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* {{Citation | last1=Rogers | first1=Hartley | title=The Theory of Recursive Functions and Effective Computability | origyear=1967 | publisher=First MIT press paperback edition | isbn=978-0-262-68052-3 | year=1987}}
 
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[[Category:Descriptive set theory]]
 
[[Category:Mathematical logic hierarchies]]
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Revision as of 14:59, 29 January 2014

In the mathematical field of descriptive set theory, a subset A of a Polish space X is projective if it is Σn1 for some positive integer n. Here A is

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 the real 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 Σn1 subset of Baire space is Σn1. It is true, however, that if a subset X of Baire space is Σn1 then there is a set of natural numbers A such that X is Σn1,A. A similar statement holds for Πn1 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.

References

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