In the mathematical field of descriptive set theory, a subset of a Polish space is projective if it is for some positive integer . Here is
- if is analytic
- if the complement of , , is
- if there is a Polish space and a subset such that is the projection of ; that is,
The choice of the Polish space 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 and ) and the projective hierarchy on subsets of Baire space (denoted by boldface letters and ). Not every subset of Baire space is . It is true, however, that if a subset X of Baire space is then there is a set of natural numbers A such that X is . A similar statement holds for 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 | ||
Σ^{0} _{0} = Π^{0} _{0} = Δ^{0} _{0} (sometimes the same as Δ^{0} _{1}) |
Σ^{0} _{0} = Π^{0} _{0} = Δ^{0} _{0} (if defined) | ||
Δ^{0} _{1} = recursive |
Δ^{0} _{1} = clopen | ||
Σ^{0} _{1} = recursively enumerable |
Π^{0} _{1} = co-recursively enumerable |
Σ^{0} _{1} = G = open |
Π^{0} _{1} = F = closed |
Δ^{0} _{2} |
Δ^{0} _{2} | ||
Σ^{0} _{2} |
Π^{0} _{2} |
Σ^{0} _{2} = F_{σ} |
Π^{0} _{2} = G_{δ} |
Δ^{0} _{3} |
Δ^{0} _{3} | ||
Σ^{0} _{3} |
Π^{0} _{3} |
Σ^{0} _{3} = G_{δσ} |
Π^{0} _{3} = F_{σδ} |
⋮ | ⋮ | ||
Σ^{0} _{<ω} = Π^{0} _{<ω} = Δ^{0} _{<ω} = Σ^{1} _{0} = Π^{1} _{0} = Δ^{1} _{0} = arithmetical |
Σ^{0} _{<ω} = Π^{0} _{<ω} = Δ^{0} _{<ω} = Σ^{1} _{0} = Π^{1} _{0} = Δ^{1} _{0} = boldface arithmetical | ||
⋮ | ⋮ | ||
Δ^{0} _{α} (α recursive) |
Δ^{0} _{α} (α countable) | ||
Σ^{0} _{α} |
Π^{0} _{α} |
Σ^{0} _{α} |
Π^{0} _{α} |
⋮ | ⋮ | ||
Σ^{0} _{ωCK1} = Π^{0} _{ωCK1} = Δ^{0} _{ωCK1} = Δ^{1} _{1} = hyperarithmetical |
Σ^{0} _{ω1} = Π^{0} _{ω1} = Δ^{0} _{ω1} = Δ^{1} _{1} = B = Borel | ||
Σ^{1} _{1} = lightface analytic |
Π^{1} _{1} = lightface coanalytic |
Σ^{1} _{1} = A = analytic |
Π^{1} _{1} = CA = coanalytic |
Δ^{1} _{2} |
Δ^{1} _{2} | ||
Σ^{1} _{2} |
Π^{1} _{2} |
Σ^{1} _{2} = PCA |
Π^{1} _{2} = CPCA |
Δ^{1} _{3} |
Δ^{1} _{3} | ||
Σ^{1} _{3} |
Π^{1} _{3} |
Σ^{1} _{3} = PCPCA |
Π^{1} _{3} = CPCPCA |
⋮ | ⋮ | ||
Σ^{1} _{<ω} = Π^{1} _{<ω} = Δ^{1} _{<ω} = Σ^{2} _{0} = Π^{2} _{0} = Δ^{2} _{0} = analytical |
Σ^{1} _{<ω} = Π^{1} _{<ω} = Δ^{1} _{<ω} = Σ^{2} _{0} = Π^{2} _{0} = Δ^{2} _{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