# Descriptive set theory

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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.

Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics. It bears close connections to metamathematics, the foundations of mathematics, and theoretical computer science. The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems.

SET: Simple definition for set:Set is a well-defined|collection of distinct objects.

In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set R of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one. It can be thought of as a vector space, a metric space, a topological space, a measure space, or a linear continuum.

## Polish spaces

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

In mathematics, a Borel set is any set in a topological space that can be formed from open sets through the operations of countable union, countable intersection, and relative complement. Borel sets are named after Émile Borel.

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 ${\displaystyle \mathbb {R} }$, the Baire space ${\displaystyle {\mathcal {N}}}$, the Cantor space ${\displaystyle {\mathcal {C}}}$, and the Hilbert cube ${\displaystyle I^{\mathbb {N} }}$.

In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named because they were first extensively studied by Polish topologists and logicians—Sierpiński, Kuratowski, Tarski and others. However, Polish spaces are mostly studied today because they are the primary setting for descriptive set theory, including the study of Borel equivalence relations. Polish spaces are also a convenient setting for more advanced measure theory, in particular in probability theory.

In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods. The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence. Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. Being so general, topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics. The branch of mathematics that studies topological spaces in their own right is called point-set topology or general topology.

In set theory, the Baire space is the set of all infinite sequences of natural numbers with a certain topology. This space is commonly used in descriptive set theory, to the extent that its elements are often called “reals.” It is denoted B, NN, ωω, ωω, or .

### Universality properties

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 is homeomorphic to a Gδ subspace of the Hilbert cube, and every Gδ 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.

In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a topology induced from that of X called the subspace topology.

In mathematics, the Hilbert cube, named after David Hilbert, is a topological space that provides an instructive example of some ideas in topology. Furthermore, many interesting topological spaces can be embedded in the Hilbert cube; that is, can be viewed as subspaces of the Hilbert cube.

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

## Borel sets

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:

In mathematical analysis and in probability theory, a σ-algebra on a set X is a collection Σ of subsets of X that includes X itself, is closed under complement, and is closed under countable unions.

• Every open subset of X is a Borel set.
• If A is a Borel set, so is ${\displaystyle X\setminus A}$. That is, the class of Borel sets are closed under complementation.
• If An is a Borel set for each natural number n, 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 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.

### Borel hierarchy

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 ${\displaystyle \mathbf {\Sigma } _{\alpha }^{0}}$, ${\displaystyle \mathbf {\Pi } _{\alpha }^{0}}$, 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 set A is declared to be ${\displaystyle \mathbf {\Sigma } _{\delta }^{0}}$, δ > 1, if there is a sequence Ai of sets, each of which is ${\displaystyle \mathbf {\Pi } _{\lambda (i)}^{0}}$ for some λ(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 all α > β. Thus the hierarchy has the following structure, where arrows indicate inclusion.

${\displaystyle {\begin{matrix}&&\mathbf {\Sigma } _{1}^{0}&&&&\mathbf {\Sigma } _{2}^{0}&&\cdots \\&\nearrow &&\searrow &&\nearrow \\\mathbf {\Delta } _{1}^{0}&&&&\mathbf {\Delta } _{2}^{0}&&&&\cdots \\&\searrow &&\nearrow &&\searrow \\&&\mathbf {\Pi } _{1}^{0}&&&&\mathbf {\Pi } _{2}^{0}&&\cdots \end{matrix}}{\begin{matrix}&&\mathbf {\Sigma } _{\alpha }^{0}&&&\cdots \\&\nearrow &&\searrow \\\quad \mathbf {\Delta } _{\alpha }^{0}&&&&\mathbf {\Delta } _{\alpha +1}^{0}&\cdots \\&\searrow &&\nearrow \\&&\mathbf {\Pi } _{\alpha }^{0}&&&\cdots \end{matrix}}}$

### Regularity properties of Borel sets

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.

## Analytic and coanalytic sets

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.

## Projective sets and Wadge degrees

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:

• 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 set A is ${\displaystyle \mathbf {\Sigma } _{n+1}^{1}}$ if there is a ${\displaystyle \mathbf {\Pi } _{n}^{1}}$ subset B of ${\displaystyle X\times X}$ such that A is the projection of B to the first coordinate.
• A set A is ${\displaystyle \mathbf {\Pi } _{n+1}^{1}}$ if there is a ${\displaystyle \mathbf {\Sigma } _{n}^{1}}$ subset B of ${\displaystyle X\times X}$ such that A is the projection of B 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 each n, 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 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.

## Borel equivalence relations

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 ${\displaystyle X\times X}$ that is an equivalence relation on X.

## Effective descriptive set theory

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.

## Table

 Lightface Boldface Σ00 = Π00 = Δ00 (sometimes the same as Δ01) Σ00 = Π00 = Δ00 (if defined) Δ01 = recursive Δ01 = clopen Σ01 = recursively enumerable Π01 = co-recursively enumerable Σ01 = G = open Π01 = F = closed Δ02 Δ02 Σ02 Π02 Σ02 = Fσ Π02 = Gδ Δ03 Δ03 Σ03 Π03 Σ03 = Gδσ Π03 = Fσδ ⋮ ⋮ Σ0<ω = Π0<ω = Δ0<ω = Σ10 = Π10 = Δ10 = arithmetical Σ0<ω = Π0<ω = Δ0<ω = Σ10 = Π10 = Δ10 = boldface arithmetical ⋮ ⋮ Δ0α (α recursive) Δ0α (α countable) Σ0α Π0α Σ0α Π0α ⋮ ⋮ Σ0 ωCK1 = Π0 ωCK1 = Δ0 ωCK1 = Δ11 = hyperarithmetical Σ0ω1 = Π0ω1 = Δ0ω1 = Δ11 = B = Borel Σ11 = lightface analytic Π11 = lightface coanalytic Σ11 = A = analytic Π11 = CA = coanalytic Δ12 Δ12 Σ12 Π12 Σ12 = PCA Π12 = CPCA Δ13 Δ13 Σ13 Π13 Σ13 = PCPCA Π13 = CPCPCA ⋮ ⋮ Σ1<ω = Π1<ω = Δ1<ω = Σ20 = Π20 = Δ20 = analytical Σ1<ω = Π1<ω = Δ1<ω = Σ20 = Π20 = Δ20 = P = projective ⋮ ⋮

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## References

• Kechris, Alexander S. (1994). Classical Descriptive Set Theory. Springer-Verlag. ISBN   0-387-94374-9.
• Moschovakis, Yiannis N. (1980). Descriptive Set Theory. North Holland. p. 2. ISBN   0-444-70199-0.