Projective hierarchy

A subset $A$ of a Polish space $X$ is projective if it is ${\boldsymbol {\Sigma }}_{n}^{1}$ for some positive integer $n$. Here $A$ is

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

Note that 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.