In mathematical logic, descriptive set theory (DST) is the study of certain classes of "well-behaved" subsets of the real line and other Polish spaces. As well as being one of the primary areas of research in set theory, it has applications to other areas of mathematics such as functional analysis, ergodic theory, the study of operator algebras and group actions, and mathematical logic.
Descriptive set theory begins with the study of Polish spaces and their Borel sets.
A Polish space is a second-countable topological space that is metrizable with a complete metric. Heuristically, it is a complete separable metric space whose metric has been "forgotten". Examples include the real line , the Baire space , the Cantor space , and the Hilbert cube .
The class of Polish spaces has several universality properties, which show that there is no loss of generality in considering Polish spaces of certain restricted forms.
Because of these universality properties, and because the Baire space has the convenient property that it is homeomorphic to , many results in descriptive set theory are proved in the context of Baire space alone.
The class of Borel sets of a topological space X consists of all sets in the smallest σ-algebra containing the open sets of X. This means that the Borel sets of X are the smallest collection of sets such that:
A fundamental result shows that any two uncountable Polish spaces X and Y are Borel isomorphic: there is a bijection from X to Y such that the preimage of any Borel set is Borel, and the image of any Borel set is Borel. This gives additional justification to the practice of restricting attention to Baire space and Cantor space, since these and any other Polish spaces are all isomorphic at the level of Borel sets.
Each Borel set of a Polish space is classified in the Borel hierarchy based on how many times the operations of countable union and complementation must be used to obtain the set, beginning from open sets. The classification is in terms of countable ordinal numbers. For each nonzero countable ordinal α there are classes , , and .
A theorem shows that any set that is or is , and any set is both and for all α > β. Thus the hierarchy has the following structure, where arrows indicate inclusion.
Classical descriptive set theory includes the study of regularity properties of Borel sets. For example, all Borel sets of a Polish space have the property of Baire and the perfect set property. Modern descriptive set theory includes the study of the ways in which these results generalize, or fail to generalize, to other classes of subsets of Polish spaces.
Just beyond the Borel sets in complexity are the analytic sets and coanalytic sets. A subset of a Polish space X is analytic if it is the continuous image of a Borel subset of some other Polish space. Although any continuous preimage of a Borel set is Borel, not all analytic sets are Borel sets. A set is coanalytic if its complement is analytic.
Many questions in descriptive set theory ultimately depend upon set-theoretic considerations and the properties of ordinal and cardinal numbers. This phenomenon is particularly apparent in the projective sets. These are defined via the projective hierarchy on a Polish space X:
As with the Borel hierarchy, for each n, any set is both and
The properties of the projective sets are not completely determined by ZFC. Under the assumption V = L, not all projective sets have the perfect set property or the property of Baire. However, under the assumption of projective determinacy, all projective sets have both the perfect set property and the property of Baire. This is related to the fact that ZFC proves Borel determinacy, but not projective determinacy.
More generally, the entire collection of sets of elements of a Polish space X can be grouped into equivalence classes, known as Wadge degrees, that generalize the projective hierarchy. These degrees are ordered in the Wadge hierarchy. The axiom of determinacy implies that the Wadge hierarchy on any Polish space is well-founded and of length Θ, with structure extending the projective hierarchy.
A contemporary area of research in descriptive set theory studies Borel equivalence relations. A Borel equivalence relation on a Polish space X is a Borel subset of that is an equivalence relation on X.
The area of effective descriptive set theory combines the methods of descriptive set theory with those of generalized recursion theory (especially hyperarithmetical theory). In particular, it focuses on lightface analogues of hierarchies of classical descriptive set theory. Thus the hyperarithmetic hierarchy is studied instead of the Borel hierarchy, and the analytical hierarchy instead of the projective hierarchy. This research is related to weaker versions of set theory such as Kripke–Platek set theory and second-order arithmetic.
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 | ||
⋮ | ⋮ |