Definitions

# Projective hierarchy

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

• $boldsymbol\left\{Sigma\right\}^1_1$ if $A$ is analytic
• $boldsymbol\left\{Pi\right\}^1_n$ if the complement of $A$, $Xsetminus A$, is $boldsymbol\left\{Sigma\right\}^1_n$
• $boldsymbol\left\{Sigma\right\}^1_\left\{n+1\right\}$ if there is a Polish space $Y$ and a $boldsymbol\left\{Pi\right\}^1_n$ subset $Csubseteq Xtimes Y$ such that $A$ is the projection of $C$; that is, $A=\left\{xin X|\left(exists yin Y\right)\left\{langle\right\}x,y\left\{rangle\right\}in C\right\}$

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.

## Relationship to the analytical hierarchy

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 $boldsymbol\left\{Sigma\right\}^1_n$ subset of Baire space is $Sigma^1_n$. It is true, however, that if a subset X of Baire space is $boldsymbol\left\{Sigma\right\}^1_n$ then there is a set of natural numbers A such that X is $Sigma^\left\{1,A\right\}_n$. A similar statement holds for $boldsymbol\left\{Pi\right\}^1_n$ 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.

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.

## References

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