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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
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| * <math>\boldsymbol{\Sigma}^1_1</math> if <math>A</math> is [[analytic set|analytic]]
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| * <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>
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| * <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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| == Relationship to the analytical hierarchy ==
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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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| 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]].
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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]]
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| [[Category:Mathematical logic hierarchies]]
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