Projective hierarchy

In the mathematical field of descriptive set theory, a subset ${\displaystyle A}$ of a Polish space ${\displaystyle X}$ is projective if it is ${\displaystyle {\boldsymbol {\Sigma }}_{n}^{1}}$ for some positive integer ${\displaystyle n}$. Here ${\displaystyle A}$ is

• ${\displaystyle {\boldsymbol {\Sigma }}_{1}^{1}}$ if ${\displaystyle A}$ is analytic
• ${\displaystyle {\boldsymbol {\Pi }}_{n}^{1}}$ if the complement of ${\displaystyle A}$, ${\displaystyle X\setminus A}$, is ${\displaystyle {\boldsymbol {\Sigma }}_{n}^{1}}$
• ${\displaystyle {\boldsymbol {\Sigma }}_{n+1}^{1}}$ if there is a Polish space ${\displaystyle Y}$ and a ${\displaystyle {\boldsymbol {\Pi }}_{n}^{1}}$ subset ${\displaystyle C\subseteq X\times Y}$ such that ${\displaystyle A}$ is the projection of ${\displaystyle C}$ onto ${\displaystyle X}$; that is, ${\displaystyle A=\{x\in X\mid \exists y\in Y:(x,y)\in C\}.}$

The choice of the Polish space ${\displaystyle Y}$ 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.