Projective hierarchy

In the mathematical field of descriptive set theory, 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, $A=\{x\in X \mid \exists y\in Y (x,y)\in C\}$

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 (denoted by lightface letters $\Sigma$ and $\Pi$ ) and the projective hierarchy on subsets of Baire space (denoted by boldface letters $\boldsymbol{\Sigma}$ and $\boldsymbol{\Pi}$ ). Not every ${\boldsymbol {\Sigma }}_{n}^{1}$ subset of Baire space is $\Sigma^1_n$ . It is true, however, that if a subset X of Baire space is ${\boldsymbol {\Sigma }}_{n}^{1}$ then there is a set of natural numbers A such that X is $\Sigma _{n}^{{1,A}}$ . A similar statement holds for ${\boldsymbol {\Pi }}_{n}^{1}$ 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.

Table

Lightface		Boldface
Σ0 0 = Π0 0 = Δ0 0 (sometimes the same as Δ0 1)		Σ0 0 = Π0 0 = Δ0 0 (if defined)
Δ0 1 = recursive		Δ0 1 = clopen
Σ0 1 = recursively enumerable	Π0 1 = co-recursively enumerable	Σ0 1 = G = open	Π0 1 = F = closed
Δ0 2		Δ0 2
Σ0 2	Π0 2	Σ0 2 = F_σ	Π0 2 = G_δ
Δ0 3		Δ0 3
Σ0 3	Π0 3	Σ0 3 = G_δσ	Π0 3 = F_σδ
...		...
Σ0 <ω = Π0 <ω = Δ0 <ω = Σ1 0 = Π1 0 = Δ1 0 = arithmetical		Σ0 <ω = Π0 <ω = Δ0 <ω = Σ1 0 = Π1 0 = Δ1 0 = boldface arithmetical
...		...
Δ0 α (α recursive)		Δ0 α (α countable)
Σ0 α	Π0 α	Σ0 α	Π0 α
...		...
Σ0 ωCK 1 = Π0 ωCK 1 = Δ0 ωCK 1 = Δ1 1 = hyperarithmetical		Σ0 ω₁ = Π0 ω₁ = Δ0 ω₁ = Δ1 1 = B = Borel
Σ1 1 = lightface analytic	Π1 1 = lightface coanalytic	Σ1 1 = A = analytic	Π1 1 = CA = coanalytic
Δ1 2		Δ1 2
Σ1 2	Π1 2	Σ1 2 = PCA	Π1 2 = CPCA
Δ1 3		Δ1 3
Σ1 3	Π1 3	Σ1 3 = PCPCA	Π1 3 = CPCPCA
...		...
Σ1 <ω = Π1 <ω = Δ1 <ω = Σ2 0 = Π2 0 = Δ2 0 = analytical		Σ1 <ω = Π1 <ω = Δ1 <ω = Σ2 0 = Π2 0 = Δ2 0 = P = projective
...		...

References

Kechris, A. S. (1995), Classical Descriptive Set Theory, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94374-9
Rogers, Hartley (1987) [1967], The Theory of Recursive Functions and Effective Computability, First MIT press paperback edition, ISBN 978-0-262-68052-3

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