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Olena Karlova

Publications and source records attributed to Olena Karlova.

At least 19 recordsLinked to original sources
arXiv2026

Borel 1 type mappings and the respective equi-families

We investigate classes of functions from a topological space to a metric space that are related to those of Borel class 1. Following the idea defining an equi-Baire 1 family (due to Lecomte) we define the respective equi-families of functions from the considered classes. We observe that studying of equi-families can be reduced to the exploration of a single orbit map with values in a product space. We consider the closure of equi-families with respect to the topology of pointwise convergence. Finally, we investigate functions $f\colon X\times Y\to Z$, for metric spaces $X,Y,Z$, with sections that are equi-continuous, equi-Baire~1 or have equi-generalized Lebesgue property with respect to measurable sets of class $α$. In particular, we generalize a result of Grande.

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arXiv2020

A characterization of the uniform convergence points set of some convergent sequence of functions

We characterize the uniform convergence points set of a pointwisely convergent sequence of real-valued functions defined on a perfectly normal space. We prove that if $X$ is a perfectly normal space which can be covered by a disjoint sequence of dense subsets and $A\subseteq X$, then $A$ is the set of points of the uniform convergence for some convergent sequence $(f_n)_{n\inω}$ of functions $f_n:X\to \mathbb R$ if and only if $A$ is $G_δ$-set which contains all isolated points of $X$. This result generalizes a theorem of Ján Borsík published in 2019.

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arXiv2019

A generalization of a Baire theorem concerning barely continuous functions

We prove that if $X$ is a paracompact space, $Y$ is a metric space and $f:X\to Y$ is a functionally fragmented map, then (i) $f$ is $σ$-discrete and functionally $F_σ$-measurable; (ii) $f$ is a Baire-one function, if $Y$ is weak adhesive and weak locally adhesive for $X$; (iii) $f$ is countably functionally fragmented, if $X$ is Lindelöff. This result generalizes one theorem of Rene Baire on classification of barely continuous functions.

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arXiv2018

Extension of fragmented Baire-one functions on Lindelöf spaces

We investigate the possibility of extension of fragmented functions from Lindelöf subspaces of completely regular spaces and find necessary and sufficient conditions on a fragmented Baire-one function to be extendable on any completely regular superspace

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arXiv2017

Diagonals of separately continuous maps with values in box products

We prove that if $X$ is a paracompact connected space and $Z=\prod_{s\in S}Z_s$ is a product of a family of equiconnected metrizable spaces endowed with the box topology, then for every Baire-one map $g:X\to Z$ there exists a separately continuous map $f:X^2\to Z$ such that $f(x,x)=g(x)$ for all $x\in X$.

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arXiv2016

Extending Baire-one functions on compact spaces

We answer a question of O. Kalenda and J. Spurný and give an example of a completely regular hereditarily Baire space $X$ and a Baire-one function $f:X\to [0,1]$ which can not be extended to a Baire-one function on $βX$.

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arXiv2016

On stable Baire classes

We introduce and study adhesive spaces. Using this concept we obtain a characterization of stable Baire maps $f:X\to Y$ of the class $α$ for wide classes of topological spaces. In particular, we prove that for a topological space $X$ and a contractible space $Y$ a map $f:X\to Y$ belongs to the $n$'th stable Baire class if and only if there exist a sequence $(f_k)_{k=1}^\infty$ of continuous maps $f_k:X\to Y$ and a sequence $(F_k)_{k=1}^\infty$ of functionally ambiguous sets of the $n$'th class in $X$ such that $f|_{F_k}=f_k|_{F_k}$ for every $k$. Moreover, we show that every monotone function $f:\mathbb R\to \mathbb R$ is of the $α$'th stable Baire class if and only if it belongs to the first stable Baire class.

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arXiv2016

Limits of sequences of continuous functions depending on finitely many coordinates

We answer two questions from {\it V.Bykov, On Baire class one functions on a product space, Topol. Appl. {199} (2016) 55--62,} and prove that every Baire one function on a subspace of a countable perfectly normal product is the pointwise limit of a sequence of continuous functions, each depending on finitely many coordinates. It is proved also that a lower semicontinuous function on a subspace of a countable perfectly normal product is the pointwise limit of an increasing sequence of continuous functions, each depending on finitely many coordinates, if and only if the function has a minorant which depends on finitely many coordinates.

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arXiv2016

The cross-topology and Lebesgue triples

The cross topology $γ$ on a product of topological spaces $X$ and $Y$ is the collection of all sets $G\subseteq X\times Y$ such that the intersection of $G$ with every vertical line and every horizontal line is an open subset of either vertical or horizontal line, respectively. For spaces $X$ and $Y$ from a wide class, which includes all spaces $\mathbb R^n$, we prove that there exists a separately continuous mapping $f:X\times Y\to (X\times Y,γ)$ which is not a pointwise limit of a sequence of continuous functions. Also we prove that every separately continuous mapping is a pointwise limit of a sequence of continuous mappings, if it is defined on the product of a strongly zero-dimensional metrizable and a topological space and acts into a topological space.

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arXiv2015

Diagonals of separately absolutely continuous mappings and their analogues

We prove that, for an interval $X\subseteq \mathbb R$ and a normed space $Z$ diagonals of separately absolute continuous mappings $f:X^2\to Z$ are exactly such mappings \mbox{$g:X\to Z$} that there is a sequence $(g_n)_{n=1}^{\infty}$ of continuous mappings $g_n:X\to Z$ with $\lim\limits_{n\to\infty}g_n(x)=g(x)$ and \mbox{$\sum\limits_{n=1}^{\infty}\|g_{n+1}(x)-g_n(x)\|<\infty$} for every $x\in X$.

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arXiv2015

On strongly separately continuous functions on sequence spaces

We study strongly separately continuous real-valued function defined on the Banach spaces $\ell_p$. Determining sets for the class of strongly separately continuous functions on $\ell_p$ are characterized. We prove that for every $1\le α<ω_1$ there exists a strongly separately continuous function which belongs the $(α+1)$'th Baire class and does not belong to the $α$'th Baire class on $\ell_p$. We show that any open set in $\ell_p$ is the set of discontinuities of a strongly separately continuous real-valued function.

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arXiv2015

On composition of Baire functions

We study the maps between topological spaces whose composition with Baire class $α$ maps also belongs to the $α$'th Baire class and give characterizations of such maps

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