Decomposing Borel functions using the Shore-Slaman join theorem

Takayuki Kihara

Displaying similar documents to “Decomposing Borel functions using the Shore-Slaman join theorem”

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...

Infinite-Dimensionality modulo Absolute Borel Classes

Vitalij Chatyrko, Yasunao Hattori (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

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For each ordinal 1 ≤ α < ω₁ we present separable metrizable spaces $X_{α}$ , $Y_{α}$ and $Z_{α}$ such that (i) $f X_{α}, f Y_{α}, f Z_{α} = ω ₀$ , where f is either trdef or ₀-trsur, (ii) $A (α) - t r i n d X_{α} = \infty$ and $M (α) - t r i n d X_{α} = - 1$ , (iii) $A (α) - t r i n d Y_{α} = - 1$ and $M (α) - t r i n d Y_{α} = \infty$ , and (iv) $A (α) - t r i n d Z_{α} = M (α) - t r i n d Z_{α} = \infty$ and $A (α + 1) \cap M (α + 1) - t r i n d Z_{α} = - 1$ . We also show that there exists no separable metrizable space $W_{α}$ with $A (α) - t r i n d W_{α} \neq \infty$ , $M (α) - t r i n d W_{α} \neq \infty$ and $A (α) \cap M (α) - t r i n d W_{α} = \infty$ , where A(α) (resp. M(α)) is the absolutely additive (resp. multiplicative) Borel class.

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....

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 $α \in [1, ω_{1})$ , 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...

Diagonals of separately continuous functions of $n$ variables with values in strongly $σ$ -metrizable spaces

Olena Karlova, Volodymyr Mykhaylyuk, Oleksandr Sobchuk (2016)

Commentationes Mathematicae Universitatis Carolinae

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We prove the result on Baire classification of mappings $f : X \times Y \to Z$ which are continuous with respect to the first variable and belongs to a Baire class with respect to the second one, where $X$ is a $P P$ -space, $Y$ is a topological space and $Z$ is a strongly $σ$ -metrizable space with additional properties. We show that for any topological space $X$ , special equiconnected space $Z$ and a mapping $g : X \to Z$ of the $(n - 1)$ -th Baire class there exists a strongly separately continuous mapping $f : X^{n} \to Z$ with the diagonal $g$ . For wide classes...

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 $f : ℝ \to ℝ$ is of the first Baire class if and only if for each $ϵ > 0$ there is a sequence of closed sets ${C_{n}}_{n = 1}^{\infty}$ such that $D_{f} = ⋃_{n = 1}^{\infty} C_{n}$ and $ω_{f} (C_{n}) < ϵ$ for each $n$ where $ω_{f} (C_{n}) = sup {| f (x) - f (y) | : x, y \in C_{n}}$ 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...

Remarks on WDC sets

Dušan Pokorný, Luděk Zajíček (2021)

Commentationes Mathematicae Universitatis Carolinae

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We study WDC sets, which form a substantial generalization of sets with positive reach and still admit the definition of curvature measures. Main results concern WDC sets $A \subset ℝ^{2}$ . We prove that, for such $A$ , the distance function $d_{A} = dist (\cdot, A)$ is a “DC aura” for $A$ , which implies that each closed locally WDC set in $ℝ^{2}$ is a WDC set. Another consequence is that compact WDC subsets of $ℝ^{2}$ form a Borel subset of the space of all compact sets.

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 $ℝ^{n}$ (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...

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.

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. ...

Complete pairs of coanalytic sets

Jean Saint Raymond (2007)

Fundamenta Mathematicae

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Let X be a Polish space, and let C₀ and C₁ be disjoint coanalytic subsets of X. The pair (C₀,C₁) is said to be complete if for every pair (D₀,D₁) of disjoint coanalytic subsets of $ω^{ω}$ there exists a continuous function $f : ω^{ω} \to X$ such that $f^{- 1} (C ₀) = D ₀$ and $f^{- 1} (C ₁) = D ₁$ . We give several explicit examples of complete pairs of coanalytic sets.

Selivanovski hard sets are hard

Janusz Pawlikowski (2015)

Fundamenta Mathematicae

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Let $H \subseteq Z \subseteq 2^{ω}$ . For n ≥ 2, we prove that if Selivanovski measurable functions from $2^{ω}$ to Z give as preimages of H all Σₙ¹ subsets of $2^{ω}$ , then so do continuous injections.

Exponential separability is preserved by some products

Vladimir Vladimirovich Tkachuk (2022)

Commentationes Mathematicae Universitatis Carolinae

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We show that exponential separability is an inverse invariant of closed maps with countably compact exponentially separable fibers. This implies that it is preserved by products with a scattered compact factor and in the products of sequential countably compact spaces. We also provide an example of a $σ$ -compact crowded space in which all countable subspaces are scattered. If $X$ is a Lindelöf space and every $Y \subset X$ with $| Y | \leq 2^{ω_{1}}$ is scattered, then $X$ is functionally countable; if every $Y \subset X$ with $| Y | \leq 2^{𝔠}$ is scattered,...

Spaces with property $(D C (ω_{1}))$

Wei-Feng Xuan, Wei-Xue Shi (2017)

Commentationes Mathematicae Universitatis Carolinae

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We prove that if $X$ is a first countable space with property $(D C (ω_{1}))$ and with a $G_{δ}$ -diagonal then the cardinality of $X$ is at most $𝔠$ . We also show that if $X$ is a first countable, DCCC, normal space then the extent of $X$ is at most $𝔠$ .

On subcompactness and countable subcompactness of metrizable spaces in ZF

Kyriakos Keremedis (2022)

Commentationes Mathematicae Universitatis Carolinae

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We show in ZF that: (i) Every subcompact metrizable space is completely metrizable, and every completely metrizable space is countably subcompact. (ii) A metrizable space $𝐗 = (X, T)$ is countably compact if and only if it is countably subcompact relative to $T$ . (iii) For every metrizable space $𝐗 = (X, T)$ , the following are equivalent: (a) $𝐗$ is compact; (b) for every open filter $ℱ$ of $𝐗$ , $⋂ {\bar{F} : F \in ℱ} \neq \emptyset$ ; (c) $𝐗$ is subcompact relative to $T$ . We also show: (iv) The negation of each of the statements, (a) every countably subcompact...