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Descriptive set theoryIn mathematical logic, descriptive set theory is the study of certain classes of "well-behaved" subsets of the real lineand other Polish spaces. As well as being one of the primary areas of research in set theory, it has applications toother areas of mathematics such as functional analysis, ergodic theory, the study of operator algebras and groupactions, and mathematical logic.

Polish spacesDescriptive 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. Equivalently, it isa complete separable metric space whose metric has been "forgotten". Examples include the real line , the Bairespace , the Cantor space , and the Hilbert cube .

Universality propertiesThe class of Polish spaces has several universality properties, which show that there is no loss of generality inconsidering 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 acontinuous image of Cantor space.

Because of these universality properties, and because the Baire space has the convenient property that it ishomeomorphic to , many results in descriptive set theory are proved in the context of Baire space alone.

Borel setsThe class of Borel sets of a topological space X consists of all sets in the smallest σ-algebra containing the open setsof X. This means that the Borel sets of X are the smallest collection of sets such that:• Every open subset of X is a Borel set.• If A is a Borel set, so is . 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 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 abijection from X to Y such that the preimage of any Borel set is Borel, and the image of any Borel set is Borel. Thisgives additional justification to the practice of restricting attention to Baire space and Cantor space, since these andany other Polish spaces are all isomorphic at the level of Borel sets.

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Borel hierarchyEach Borel set of a Polish space is classified in the Borel hierarchy based on how many times the operations ofcountable union and complementation must be used to obtain the set, beginning from open sets. The classification isin terms of countable ordinal numbers. For each nonzero countable ordinal α there are classes , , and .• Every open set is declared to be .• A set is declared to be if and only if its complement is .• A set A is declared to be , δ > 1, if there is a sequence ⟨ Ai ⟩ of sets, each of which is for some λ(i) < δ,

such that .• A set is if and only if it is both 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.

Regularity properties of Borel setsClassical descriptive set theory includes the study of regularity properties of Borel sets. For example, all Borel setsof a Polish space have the property of Baire and the perfect set property. Modern descriptive set theory includes thestudy of the ways in which these results generalize, or fail to generalize, to other classes of subsets of Polish spaces.

Analytic and coanalytic setsJust beyond the Borel sets in complexity are the analytic sets and coanalytic sets. A subset of a Polish space X isanalytic if it is the continuous image of a Borel subset of some other Polish space. Although any continuouspreimage 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 degreesMany questions in descriptive set theory ultimately depend upon set-theoretic considerations and the properties ofordinal and cardinal numbers. This phenomenon is particularly apparent in the projective sets. These are defined viathe projective hierarchy on a Polish space X:

• A set is declared to be if it is analytic.• A set is if it is coanalytic.• A set A is if there is a subset B of such that A is the projection of B to the first coordinate.• A set A is if there is a subset B of such that A is the projection of B to the first coordinate.• A set is if it is both and .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 allprojective sets have the perfect set property or the property of Baire. However, under the assumption of projectivedeterminacy, all projective sets have both the perfect set property and the property of Baire. This is related to the factthat 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

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hierarchy. The axiom of determinacy implies that the Wadge hierarchy on any Polish space is well-founded and oflength Θ, with structure extending the projective hierarchy.

Borel equivalence relationsA contemporary area of research in descriptive set theory studies Borel equivalence relations. A Borel equivalencerelation on a Polish space X is a Borel subset of that is an equivalence relation on X.

Effective descriptive set theoryThe area of effective descriptive set theory combines the methods of descriptive set theory with those of generalizedrecursion theory (especially hyperarithmetical theory). In particular, it focuses on lightface analogues of hierarchiesof classical descriptive set theory. Thus the hyperarithmetic hierarchy is studied instead of the Borel hierarchy, andthe analytical hierarchy instead of the projective hierarchy. This research is related to weaker version of set theorysuch as Kripke-Platek set theory and second-order arithmetic.

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. ISBN 0-444-70199-0. Second edition

available online [1]

External links• Descriptive set theory [2], David Marker, 2002. Lecture notes.

References[1] http:/ / www. math. ucla. edu/ ~ynm/ books. htm[2] http:/ / www. math. uic. edu/ ~marker/ math512/ dst. pdf

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Article Sources and ContributorsDescriptive set theory  Source: http://en.wikipedia.org/w/index.php?oldid=540537188  Contributors: Blotwell, Brad7777, CBM, Charles Matthews, Dysprosia, Gala.martin, Giftlite, HenryDelforn (old), JerroldPease-Atlanta, JohnBlackburne, Maximus Rex, Mrw7, Msh210, Pascal.Tesson, Perelaar, Revolver, Stotr, Tkuvho, Tosha, Trovatore, 24 anonymous edits

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