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Displaying similar documents to “On Borel reducibility in generalized Baire space”

Consistency of the Silver dichotomy in generalised Baire space

Sy-David Friedman (2014)

Fundamenta Mathematicae

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Silver’s fundamental dichotomy in the classical theory of Borel reducibility states that any Borel (or even co-analytic) equivalence relation with uncountably many classes has a perfect set of classes. The natural generalisation of this to the generalised Baire space κ κ for a regular uncountable κ fails in Gödel’s L, even for κ-Borel equivalence relations. We show here that Silver’s dichotomy for κ-Borel equivalence relations in κ κ for uncountable regular κ is however consistent (with...

On the closure of Baire classes under transfinite convergences

Tamás Mátrai (2004)

Fundamenta Mathematicae

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Let X be a Polish space and Y be a separable metric space. For a fixed ξ < ω₁, consider a family f α : X Y ( α < ω ) of Baire-ξ functions. Answering a question of Tomasz Natkaniec, we show that if for a function f: X → Y, the set α < ω : f α ( x ) f ( x ) is finite for every x ∈ X, then f itself is necessarily Baire-ξ. The proof is based on a characterization of Σ η sets which can be interesting in its own right.

Functions of Baire class one

Denny H. Leung, Wee-Kee Tang (2003)

Fundamenta Mathematicae

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Let K be a compact metric space. A real-valued function on K is said to be of Baire class one (Baire-1) if it is the pointwise limit of a sequence of continuous functions. We study two well known ordinal indices of Baire-1 functions, the oscillation index β and the convergence index γ. It is shown that these two indices are fully compatible in the following sense: a Baire-1 function f satisfies β ( f ) ω ξ · ω ξ for some countable ordinals ξ₁ and ξ₂ if and only if there exists a sequence (fₙ) of Baire-1...

Extension of functions with small oscillation

Denny H. Leung, Wee-Kee Tang (2006)

Fundamenta Mathematicae

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A classical theorem of Kuratowski says that every Baire one function on a G δ subspace of a Polish (= separable completely metrizable) space X can be extended to a Baire one function on X. Kechris and Louveau introduced a finer gradation of Baire one functions into small Baire classes. A Baire one function f is assigned into a class in this hierarchy depending on its oscillation index β(f). We prove a refinement of Kuratowski’s theorem: if Y is a subspace of a metric space X and f is a...

Rudin-like sets and hereditary families of compact sets

Étienne Matheron, Miroslav Zelený (2005)

Fundamenta Mathematicae

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We show that a comeager Π₁¹ hereditary family of compact sets must have a dense G δ subfamily which is also hereditary. Using this, we prove an “abstract” result which implies the existence of independent ℳ ₀-sets, the meagerness of ₀-sets with the property of Baire, and generalizations of some classical results of Mycielski. Finally, we also give some natural examples of true F σ δ sets.

Insertion of a Contra-Baire- 1 (Baire- . 5 ) Function

Majid Mirmiran (2019)

Communications in Mathematics

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Necessary and sufficient conditions in terms of lower cut sets are given for the insertion of a Baire- . 5 function between two comparable real-valued functions on the topological spaces that F σ -kernel of sets are F σ -sets.

Decomposing Borel functions using the Shore-Slaman join theorem

Takayuki Kihara (2015)

Fundamenta Mathematicae

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Jayne and Rogers proved that every function from an analytic space into a separable metrizable space is decomposable into countably many continuous functions with closed domains if and only if the preimage of each F σ set under that function is again F σ . Many researchers conjectured that the Jayne-Rogers theorem can be generalized to all finite levels of Borel functions. In this paper, by using the Shore-Slaman join theorem on the Turing degrees, we show the following variant of the Jayne-Rogers...

Distances to spaces of affine Baire-one functions

Jiří Spurný (2010)

Studia Mathematica

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Let E be a Banach space and let ( B E * ) and ( B E * ) denote the space of all Baire-one and affine Baire-one functions on the dual unit ball B E * , respectively. We show that there exists a separable L₁-predual E such that there is no quantitative relation between d i s t ( f , ( B E * ) ) and d i s t ( f , ( B E * ) ) , where f is an affine function on B E * . If the Banach space E satisfies some additional assumption, we prove the existence of some such dependence.

Operators on the stopping time space

Dimitris Apatsidis (2015)

Studia Mathematica

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Let S¹ be the stopping time space and ℬ₁(S¹) be the Baire-1 elements of the second dual of S¹. To each element x** in ℬ₁(S¹) we associate a positive Borel measure μ x * * on the Cantor set. We use the measures μ x * * : x * * ( S ¹ ) to characterize the operators T: X → S¹, defined on a space X with an unconditional basis, which preserve a copy of S¹. In particular, if X = S¹, we show that T preserves a copy of S¹ if and only if μ T * * ( x * * ) : x * * ( S ¹ ) is non-separable as a subset of ( 2 ) .

Descriptive properties of elements of biduals of Banach spaces

Pavel Ludvík, Jiří Spurný (2012)

Studia Mathematica

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If E is a Banach space, any element x** in its bidual E** is an affine function on the dual unit ball B E * that might possess a variety of descriptive properties with respect to the weak* topology. We prove several results showing that descriptive properties of x** are quite often determined by the behaviour of x** on the set of extreme points of B E * , generalizing thus results of J. Saint Raymond and F. Jellett. We also prove a result on the relation between Baire classes and intrinsic Baire...

Borel classes of uniformizations of sets with large sections

Petr Holický (2010)

Fundamenta Mathematicae

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We give several refinements of known theorems on Borel uniformizations of sets with “large sections”. In particular, we show that a set B ⊂ [0,1] × [0,1] which belongs to Σ α , α ≥ 2, and which has all “vertical” sections of positive Lebesgue measure, has a Π α uniformization which is the graph of a Σ α -measurable mapping. We get a similar result for sets with nonmeager sections. As a corollary we derive an improvement of Srivastava’s theorem on uniformizations for Borel sets with G δ sections. ...

Baire one functions and their sets of discontinuity

Jonald P. Fenecios, Emmanuel A. Cabral, Abraham P. Racca (2016)

Mathematica Bohemica

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A characterization of functions in the first Baire class in terms of their sets of discontinuity is given. More precisely, a function f : is of the first Baire class if and only if for each ϵ > 0 there is a sequence of closed sets { C n } n = 1 such that D f = n = 1 C n and ω f ( C n ) < ϵ for each n where ω f ( C n ) = sup { | f ( x ) - f ( y ) | : x , y C n } and D f denotes the set of points of discontinuity of f . The proof of the main theorem is based on a recent ϵ - δ characterization of Baire class one functions as well as on a well-known theorem due to Lebesgue. Some direct applications...

A remark on functions continuous on all lines

Luděk Zajíček (2019)

Commentationes Mathematicae Universitatis Carolinae

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We prove that each linearly continuous function f on n (i.e., each function continuous on all lines) belongs to the first Baire class, which answers a problem formulated by K. C. Ciesielski and D. Miller (2016). The same result holds also for f on an arbitrary Banach space X , if f has moreover the Baire property. We also prove (extending a known finite-dimensional result) that such f on a separable X is continuous at all points outside a first category set which is also null in any usual...

Baire classes of complex L 1 -preduals

Pavel Ludvík, Jiří Spurný (2015)

Czechoslovak Mathematical Journal

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Let X be a complex L 1 -predual, non-separable in general. We investigate extendability of complex-valued bounded homogeneous Baire- α functions on the set ext B X * of the extreme points of the dual unit ball B X * to the whole unit ball B X * . As a corollary we show that, given α [ 1 , ω 1 ) , the intrinsic α -th Baire class of X can be identified with the space of bounded homogeneous Baire- α functions on the set ext B X * when ext B X * satisfies certain topological assumptions. The paper is intended to be a complex counterpart to...

Effective decomposition of σ-continuous Borel functions

Gabriel Debs (2014)

Fundamenta Mathematicae

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We prove that if a Δ¹₁ function f with Σ¹₁ domain X is σ-continuous then one can find a Δ¹₁ covering ( A ) n ω of X such that f | A is continuous for all n. This is an effective version of a recent result by Pawlikowski and Sabok, generalizing an earlier result of Solecki.

The effective Borel hierarchy

M. Vanden Boom (2007)

Fundamenta Mathematicae

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Let K be a subclass of Mod() which is closed under isomorphism. Vaught showed that K is Σ α (respectively, Π α ) in the Borel hierarchy iff K is axiomatized by an infinitary Σ α (respectively, Π α ) sentence. We prove a generalization of Vaught’s theorem for the effective Borel hierarchy, i.e. the Borel sets formed by union and complementation over c.e. sets. This result says that we can axiomatize an effective Σ α or effective Π α Borel set with a computable infinitary sentence of the same complexity....

On strong measure zero subsets of κ 2

Aapo Halko, Saharon Shelah (2001)

Fundamenta Mathematicae

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We study the generalized Cantor space κ 2 and the generalized Baire space κ κ as analogues of the classical Cantor and Baire spaces. We equip κ κ with the topology where a basic neighborhood of a point η is the set ν: (∀j < i)(ν(j) = η(j)), where i < κ. We define the concept of a strong measure zero set of κ 2 . We prove for successor κ = κ < κ that the ideal of strong measure zero sets of κ 2 is κ -additive, where κ is the size of the smallest unbounded family in κ κ , and that the generalized Borel...

Transportation flow problems with Radon measure variables

Marcus Wagner (2000)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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For a multidimensional control problem ( P ) K involving controls u L , we construct a dual problem ( D ) K in which the variables ν to be paired with u are taken from the measure space rca (Ω,) instead of ( L ) * . For this purpose, we add to ( P ) K a Baire class restriction for the representatives of the controls u. As main results, we prove a strong duality theorem and saddle-point conditions.

Generic power series on subsets of the unit disk

Balázs Maga, Péter Maga (2022)

Czechoslovak Mathematical Journal

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We examine the boundary behaviour of the generic power series f with coefficients chosen from a fixed bounded set Λ in the sense of Baire category. Notably, we prove that for any open subset U of the unit disk D with a nonreal boundary point on the unit circle, f ( U ) is a dense set of . As it is demonstrated, this conclusion does not necessarily hold for arbitrary open sets accumulating to the unit circle. To complement these results, a characterization of coefficient sets having this property...

F σ -mappings and the invariance of absolute Borel classes

Petr Holický, Jiří Spurný (2004)

Fundamenta Mathematicae

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It is proved that F σ -mappings preserve absolute Borel classes, which improves results of R. W. Hansell, J. E. Jayne and C. A. Rogers. The proof is based on the fact that any F σ -mapping f: X → Y of an absolute Suslin metric space X onto an absolute Suslin metric space Y becomes a piecewise perfect mapping when restricted to a suitable F σ -set X X satisfying f ( X ) = Y .

On the complexity of subspaces of S ω

Carlos Uzcátegui (2003)

Fundamenta Mathematicae

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Let (X,τ) be a countable topological space. We say that τ is an analytic (resp. Borel) topology if τ as a subset of the Cantor set 2 X (via characteristic functions) is an analytic (resp. Borel) set. For example, the topology of the Arkhangel’skiĭ-Franklin space S ω is F σ δ . In this paper we study the complexity, in the sense of the Borel hierarchy, of subspaces of S ω . We show that S ω has subspaces with topologies of arbitrarily high Borel rank and it also has subspaces with a non-Borel topology....