Borel sets with σ-compact sections for nonseparable spaces

Petr Holický

Displaying similar documents to “Borel sets with σ-compact sections for nonseparable spaces”

Borel spaces

K. P. S. Bhaskara Rao, B. V. Rao

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CONTENTSIntroduction............................................................................... 5Chapter 1. Borel spaces........................................................ 7 § 1. Borel spaces....................................................... 7 § 2. Classical descriptive set theory............................... 10 § 3. Measure and category............................................... 12 § 4. Countably generated structures.............................. 13 § 5. Product...

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 Borel reducibility in generalized Baire space

Sy-David Friedman, Tapani Hyttinen, Vadim Kulikov (2015)

Fundamenta Mathematicae

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We study the Borel reducibility of Borel equivalence relations on the generalized Baire space $κ^{κ}$ for an uncountable κ with $κ^{< κ} = κ$ . The theory looks quite different from its classical counterpart where κ = ω, although some basic theorems do generalize.

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 $Σ ⁰_{α}$ , α ≥ 2, and which has all “vertical” sections of positive Lebesgue measure, has a $Π ⁰_{α}$ uniformization which is the graph of a $Σ ⁰_{α}$ -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. ...

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....

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 ${(A ₙ)}_{n \in ω}$ of X such that $f_{| A ₙ}$ is continuous for all n. This is an effective version of a recent result by Pawlikowski and Sabok, generalizing an earlier result of Solecki.

$F_{σ}$ -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 $X_{\infty} \subset X$ satisfying $f (X_{\infty}) = Y$ .

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...

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....

Pointwise convergence and the Wadge hierarchy

Alessandro Andretta, Alberto Marcone (2001)

Commentationes Mathematicae Universitatis Carolinae

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We show that if $X$ is a $Σ_{1}^{1}$ separable metrizable space which is not $σ$ -compact then $C_{p}^{*} (X)$ , the space of bounded real-valued continuous functions on $X$ with the topology of pointwise convergence, is Borel- $Π_{1}^{1}$ -complete. Assuming projective determinacy we show that if $X$ is projective not $σ$ -compact and $n$ is least such that $X$ is $Σ_{n}^{1}$ then $C_{p} (X)$ , the space of real-valued continuous functions on $X$ with the topology of pointwise convergence, is Borel- $Π_{n}^{1}$ -complete. We also prove a simultaneous improvement of theorems...

Borel completeness of some ℵ₀-stable theories

Michael C. Laskowski, Saharon Shelah (2015)

Fundamenta Mathematicae

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We study ℵ₀-stable theories, and prove that if T either has eni-DOP or is eni-deep, then its class of countable models is Borel complete. We introduce the notion of λ-Borel completeness and prove that such theories are λ-Borel complete. Using this, we conclude that an ℵ₀-stable theory satisfies $I_{\infty, ℵ ₀} (T, λ) = 2^{λ}$ for all cardinals λ if and only if T either has eni-DOP or is eni-deep.

Extension maps in ultradifferentiable and ultraholomorphic function spaces

Jean Schmets, Manuel Valdivia (2000)

Studia Mathematica

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The problem of the existence of extension maps from 0 to ℝ in the setting of the classical ultradifferentiable function spaces has been solved by Petzsche [9] by proving a generalization of the Borel and Mityagin theorems for $C^{\infty}$ -spaces. We get a Ritt type improvement, i.e. from 0 to sectors of the Riemann surface of the function log for spaces of ultraholomorphic functions, by first establishing a generalization to some nonclassical ultradifferentiable function spaces.

Continuous linear functionals on the space of Borel vector measures

Pola Siwek (2008)

Annales Polonici Mathematici

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We study properties of the space ℳ of Borel vector measures on a compact metric space X, taking values in a Banach space E. The space ℳ is equipped with the Fortet-Mourier norm ${| | \cdot | |}_{ℱ}$ and the semivariation norm ||·||(X). The integral introduced by K. Baron and A. Lasota plays the most important role in the paper. Investigating its properties one can prove that in most cases the space $(ℳ, | | \cdot | |_{ℱ}) *$ is contained in but not equal to the space (ℳ,||·||(X))*. We obtain a representation of the continuous functionals...

A López-Escobar theorem for metric structures, and the topological Vaught conjecture

Samuel Coskey, Martino Lupini (2016)

Fundamenta Mathematicae

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We show that a version of López-Escobar’s theorem holds in the setting of model theory for metric structures. More precisely, let denote the Urysohn sphere and let Mod(,) be the space of metric -structures supported on . Then for any Iso()-invariant Borel function f: Mod(,) → [0,1], there exists a sentence ϕ of $_{ω ₁ ω}$ such that for all M ∈ Mod(,) we have $f (M) = ϕ^{M}$ . This answers a question of Ivanov and Majcher-Iwanow. We prove several consequences, for example every orbit equivalence relation of...

Banach spaces of bounded Szlenk index

E. Odell, Th. Schlumprecht, A. Zsák (2007)

Studia Mathematica

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For a countable ordinal α we denote by $_{α}$ the class of separable, reflexive Banach spaces whose Szlenk index and the Szlenk index of their dual are bounded by α. We show that each $_{α}$ admits a separable, reflexive universal space. We also show that spaces in the class $_{ω^{α \cdot ω}}$ embed into spaces of the same class with a basis. As a consequence we deduce that each $_{α}$ is analytic in the Effros-Borel structure of subspaces of C[0,1].

Applications of some results of infinite-dimensional topology to the topological classification of operator images

Taras Banakh, Tadeusz Dobrowolski, Anatoliĭ Plichko

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This volume consists of three relatively independent articles devoted to the topological study of the so-called operator images and weak unit balls of Banach spaces. These articles are: “The topological classification of weak unit balls of Banach spaces” by T. Banakh, “The topological and Borel classification of operator images” by T. Banakh, T. Dobrowolski and A. Plichko, and “Operator images homeomorphic to $Σ^{ω}$ ” by T. Banakh. The articles summarize investigations that has been done by...

Sidon sets and $I^{0}$ -sets

David Grow (1987)

Colloquium Mathematicae

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Failure of the Factor Theorem for Borel pre-Hilbert spaces

Tadeusz Dobrowolski, Witold Marciszewski (2002)

Fundamenta Mathematicae

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In every infinite-dimensional Fréchet space X, we construct a linear subspace E such that E is an $F_{σ δ σ}$ -subset of X and contains a retract R so that $R \times E^{ω}$ is not homeomorphic to $E^{ω}$ . This shows that Toruńczyk’s Factor Theorem fails in the Borel case.

A descriptive view of unitary group representations

Simon Thomas (2015)

Journal of the European Mathematical Society

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In this paper, we will study the relative complexity of the unitary duals of countable groups. In particular, we will explain that if $G$ and $H$ are countable amenable non-type I groups, then the unitary duals of $G$ and $H$ are Borel isomorphic.

Borel summable solutions of the Burgers equation

Grzegorz Łysik (2009)

Annales Polonici Mathematici

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We give necessary and sufficient conditions for the formal power series solutions to the initial value problem for the Burgers equation $\partial_{t} u - \partial_{x} ² u = \partial_{x} (u ²)$ to be convergent or Borel summable.

Filter descriptive classes of Borel functions

Gabriel Debs, Jean Saint Raymond (2009)

Fundamenta Mathematicae

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We first prove that given any analytic filter ℱ on ω the set of all functions f on $2^{ω}$ which can be represented as the pointwise limit relative to ℱ of some sequence ${(f ₙ)}_{n \in ω}$ of continuous functions ( $f = l i m_{ℱ} f ₙ$ ), is exactly the set of all Borel functions of class ξ for some countable ordinal ξ that we call the rank of ℱ. We discuss several structural properties of this rank. For example, we prove that any free Π⁰₄ filter is of rank 1.

Wijsman hyperspaces of non-separable metric spaces

Rodrigo Hernández-Gutiérrez, Paul J. Szeptycki (2015)

Fundamenta Mathematicae

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Given a metric space ⟨X,ρ⟩, consider its hyperspace of closed sets CL(X) with the Wijsman topology $τ_{W (ρ)}$ . It is known that $⟨ C L (X), τ_{W (ρ)} ⟩$ is metrizable if and only if X is separable, and it is an open question by Di Maio and Meccariello whether this is equivalent to $⟨ C L (X), τ_{W (ρ)} ⟩$ being normal. We prove that if the weight of X is a regular uncountable cardinal and X is locally separable, then $⟨ C L (X), τ_{W (ρ)} ⟩$ is not normal. We also solve some questions by Cao, Junnila and Moors regarding isolated points in Wijsman hyperspaces. ...

Hyperspaces of Finite Sets in Universal Spaces for Absolute Borel Classes

Kotaro Mine, Katsuro Sakai, Masato Yaguchi (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

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By Fin(X) (resp. $F i n^{k} (X)$ ), we denote the hyperspace of all non-empty finite subsets of X (resp. consisting of at most k points) with the Vietoris topology. Let ℓ₂(τ) be the Hilbert space with weight τ and $ℓ ₂^{f} (τ)$ the linear span of the canonical orthonormal basis of ℓ₂(τ). It is shown that if $E = ℓ ₂^{f} (τ)$ or E is an absorbing set in ℓ₂(τ) for one of the absolute Borel classes $_{α} (τ)$ and $_{α} (τ)$ of weight ≤ τ (α > 0) then Fin(E) and each $F i n^{k} (E)$ are homeomorphic to E. More generally, if X is a connected E-manifold then Fin(X)...

Median for metric spaces

Nacereddine Belili, Henri Heinich (2001)

Applicationes Mathematicae

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We consider a Köthe space $(, | | \cdot | |_{)}$ of random variables (r.v.) defined on the Lebesgue space ([0,1],B,λ). We show that for any sub-σ-algebra ℱ of B and for all r.v.’s X with values in a separable finitely compact metric space (M,d) such that d(X,x) ∈ for all x ∈ M (we then write X ∈ (M)), there exists a median of X given ℱ, i.e., an ℱ-measurable r.v. Y ∈ (M) such that ${| | d (X, Y) | |}_{\leq} | | d (X, Z) | |$ for all ℱ-measurable Z. We develop the basic theory of these medians, we show the convergence of empirical medians and we give...

Johnson's projection, Kalton's property (M*), and M-ideals of compact operators

Olav Nygaard, Märt Põldvere (2009)

Studia Mathematica

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Let X and Y be Banach spaces. We give a “non-separable” proof of the Kalton-Werner-Lima-Oja theorem that the subspace (X,X) of compact operators forms an M-ideal in the space (X,X) of all continuous linear operators from X to X if and only if X has Kalton’s property (M*) and the metric compact approximation property. Our proof is a quick consequence of two main results. First, we describe how Johnson’s projection P on (X,Y)* applies to f ∈ (X,Y)* when f is represented via a Borel (with...

On an Invariant Borel Measure in Hilbert Space

G. Pantsulaia (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

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An example of a nonzero σ-finite Borel measure μ with everywhere dense linear manifold $_{μ}$ of admissible (in the sense of invariance) translation vectors is constructed in the Hilbert space ℓ₂ such that μ and any shift $μ^{(a)}$ of μ by a vector $a \in ℓ ₂ ∖_{μ}$ are neither equivalent nor orthogonal. This extends a result established in [7].

On the tangency of sets in generalized metric spaces for certain functions of the class $F_{ρ}^{*}$

T. Konik (1991)

Matematički Vesnik

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The trilinear embedding theorem

Hitoshi Tanaka (2015)

Studia Mathematica

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Let $σ_{i}$ , i = 1,2,3, denote positive Borel measures on ℝⁿ, let denote the usual collection of dyadic cubes in ℝⁿ and let K: → [0,∞) be a map. We give a characterization of a trilinear embedding theorem, that is, of the inequality $\sum_{Q \in} K (Q) \prod_{i = 1}^{3} | \int_{Q} f_{i} d σ_{i} | \leq C \prod_{i = 1}^{3} | | f_{i} {| |}_{L^{p_{i}} (d σ_{i})}$ in terms of a discrete Wolff potential and Sawyer’s checking condition, when 1 < p₁,p₂,p₃ < ∞ and 1/p₁ + 1/p₂ + 1/p₃ ≥ 1.