Inmathematical logic,descriptive set theory (DST) is the study of certain classes of "well-behaved"subsets of thereal line and otherPolish spaces. As well as being one of the primary areas of research inset theory, it has applications to other areas of mathematics such asfunctional analysis,ergodic theory, the study ofoperator algebras andgroup actions, andmathematical logic.

Descriptive set theory begins with the study of Polish spaces and theirBorel sets.

APolish space is asecond-countabletopological space that ismetrizable with acomplete metric. Heuristically, it is a completeseparable metric space whose metric has been "forgotten". Examples include thereal line${\displaystyle \mathbb{R}}$, theBaire space${\displaystyle \mathcal{N}}$, theCantor space${\displaystyle \mathcal{C}}$, and theHilbert cube${\displaystyle I^{\mathbb{N}}}$.

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.

• Every Polish space ishomeomorphic to aG_δ subspace of theHilbert cube, and everyG_δ subspace of the Hilbert cube is Polish.
• Every Polish space is obtained as a continuous image of Baire space; in fact every Polish space is the image of a continuous bijection defined on a closed subset of Baire space. Similarly, every compact Polish space is a continuous image of Cantor space.

Because of these universality properties, and because the Baire space${\displaystyle \mathcal{N}}$ has the convenient property that it ishomeomorphic to${\displaystyle {\mathcal{N}}^{\omega }}$, many results in descriptive set theory are proved in the context of Baire space alone.

The class ofBorel sets of a topological spaceX consists of all sets in the smallestσ-algebra containing the open sets ofX. This means that the Borel sets ofX are the smallest collection of sets such that:

• Every open subset ofX is a Borel set.
• IfA is a Borel set, so is${\displaystyle X\setminus A}$. That is, the class of Borel sets are closed under complementation.
• IfA_n is a Borel set for each natural numbern, then the union${\displaystyle \bigcup A_n}$ is a Borel set. That is, the Borel sets are closed under countable unions.

A fundamental result shows that any two uncountable Polish spacesX andY areBorel isomorphic: there is a bijection fromX toY 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 theBorel 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 ofcountableordinal numbers. For each nonzero countable ordinalα there are classes${\displaystyle {\mathbf{\Sigma }}_{\alpha }^0}$,${\displaystyle {\mathbf{\Pi }}_{\alpha }^0}$, and${\displaystyle {\mathbf{\Delta }}_{\alpha }^0}$.

• Every open set is declared to be${\displaystyle {\mathbf{\Sigma }}_1^0}$.
• A set is declared to be${\displaystyle {\mathbf{\Pi }}_{\alpha }^0}$ if and only if its complement is${\displaystyle {\mathbf{\Sigma }}_{\alpha }^0}$.
• A setA is declared to be${\displaystyle {\mathbf{\Sigma }}_{\delta }^0}$,δ > 1, if there is a sequence ⟨A_i ⟩ of sets, each of which is${\displaystyle {\mathbf{\Pi }}_{\lambda (i)}^0}$ for someλ(i) <δ, such that${\displaystyle A=\bigcup A_i}$.
• A set is${\displaystyle {\mathbf{\Delta }}_{\alpha }^0}$ if and only if it is both${\displaystyle {\mathbf{\Sigma }}_{\alpha }^0}$ and${\displaystyle {\mathbf{\Pi }}_{\alpha }^0}$.

A theorem shows that any set that is${\displaystyle {\mathbf{\Sigma }}_{\alpha }^0}$ or${\displaystyle {\mathbf{\Pi }}_{\alpha }^0}$ is${\displaystyle {\mathbf{\Delta }}_{\alpha +1}^0}$, and any${\displaystyle {\mathbf{\Delta }}_{\beta }^0}$ set is both${\displaystyle {\mathbf{\Sigma }}_{\alpha }^0}$ and${\displaystyle {\mathbf{\Pi }}_{\alpha }^0}$ for allα >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 theproperty of Baire and theperfect 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 theanalytic sets andcoanalytic sets. A subset of a Polish spaceX isanalytic 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 iscoanalytic if its complement is analytic.

Many questions in descriptive set theory ultimately depend uponset-theoretic considerations and the properties ofordinal andcardinal numbers. This phenomenon is particularly apparent in theprojective sets. These are defined via theprojective hierarchy on a Polish spaceX:

• A set is declared to be${\displaystyle {\mathbf{\Sigma }}_1^1}$ if it is analytic.
• A set is${\displaystyle {\mathbf{\Pi }}_1^1}$ if it is coanalytic.
• A setA is${\displaystyle {\mathbf{\Sigma }}_{n+1}^1}$ if there is a${\displaystyle {\mathbf{\Pi }}_n^1}$ subsetB of${\displaystyle X\times X}$ such thatA is the projection ofB to the first coordinate.
• A setA is${\displaystyle {\mathbf{\Pi }}_{n+1}^1}$ if there is a${\displaystyle {\mathbf{\Sigma }}_n^1}$ subsetB of${\displaystyle X\times X}$ such thatA is the projection ofB to the first coordinate.
• A set is${\displaystyle {\mathbf{\Delta }}_n^1}$ if it is both${\displaystyle {\mathbf{\Pi }}_n^1}$ and${\displaystyle {\mathbf{\Sigma }}_n^1}$ .

As with the Borel hierarchy, for eachn, any${\displaystyle {\mathbf{\Delta }}_n^1}$ set is both${\displaystyle {\mathbf{\Sigma }}_{n+1}^1}$ and${\displaystyle {\mathbf{\Pi }}_{n+1}^1}$.

The properties of the projective sets are not completely determined by ZFC. Under the assumptionV = L, not all projective sets have the perfect set property or the property of Baire. However, under the assumption ofprojective determinacy, all projective sets have both the perfect set property and the property of Baire. This is related to the fact that ZFC provesBorel determinacy, but not projective determinacy.

There are also generic extensions of${\displaystyle L}$ for any natural number${\displaystyle n>2}$ in which${\displaystyle \mathcal{P}(\omega )\cap L}$ consists of all the lightface${\displaystyle {\mathrm{\Delta }}_n^1}$ subsets of${\displaystyle \omega }$.^[1]

More generally, the entire collection of sets of elements of a Polish spaceX can be grouped into equivalence classes, known asWadge degrees, that generalize the projective hierarchy. These degrees are ordered in theWadge hierarchy. Theaxiom of determinacy implies that the Wadge hierarchy on any Polish space is well-founded and of lengthΘ, with structure extending the projective hierarchy.

A contemporary area of research in descriptive set theory studiesBorel equivalence relations. ABorel equivalence relation on a Polish spaceX is a Borel subset of${\displaystyle X\times X}$ that is anequivalence relation onX.

The area ofeffective descriptive set theory combines the methods of descriptive set theory with those ofgeneralized recursion theory (especiallyhyperarithmetical theory). In particular, it focuses onlightface analogues of hierarchies of classical descriptive set theory. Thus thehyperarithmetic hierarchy is studied instead of the Borel hierarchy, and theanalytical hierarchy instead of the projective hierarchy. This research is related to weaker versions of set theory such asKripke–Platek set theory andsecond-order arithmetic.

• Pointclass
• Prewellordering
• Scale property

• Kechris, Alexander S. (1994).Classical Descriptive Set Theory. Springer-Verlag.ISBN 0-387-94374-9.
• Moschovakis, Yiannis N. (1980).Descriptive Set Theory. North Holland.ISBN 0-444-70199-0.
• Bartoszynski, Tomek; Judah, Haim (1995).Set Theory: On the Structure of the Real Line. Natick: A K Peters/CRC Press.ISBN 978-1-56881-044-7.

• Descriptive set theory, David Marker, 2002. Lecture notes.

• Descriptive set theory

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