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

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

The choice of the Polish space Y {\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.