Bad Wadge-like reducibilities on the Baire space

Luca Motto Ros

Displaying similar documents to “Bad Wadge-like reducibilities on the Baire space”

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...

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...

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.

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...

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...

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...

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 $ℂ$ . 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...

Degrees of compatible $L$ -subsets and compatible mappings

Fu Gui Shi, Yan Sun (2024)

Kybernetika

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Based on a completely distributive lattice $L$ , degrees of compatible $L$ -subsets and compatible mappings are introduced in an $L$ -approximation space and their characterizations are given by four kinds of cut sets of $L$ -subsets and $L$ -equivalences, respectively. Besides, some characterizations of compatible mappings and compatible degrees of mappings are given by compatible $L$ -subsets and compatible degrees of $L$ -subsets. Finally, the notion of complete $L$ -sublattices is introduced and it is shown...