Editing Projective hierarchy (section)

{{Short description|Descriptive set theory concept}}
{{redirect|Projective set|the card game|Projective Set (game)}}

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 (mathematics)|projection]] of <math>C</math> onto <math>X</math>; that is, <math>A=\{x\in X \mid \exists y\in Y : (x,y)\in C\}.</math>

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]].