arXiv subjects
Publications and source records attributed to Olena Karlova.
We generalize the Lebesgue-Hausdorff Theorem on the characterization of Baire-one functions for $σ$-strongly functionally discrete mappings defined on arbitrary topological spaces
We prove that every $C^*$-embedded subset of $ß$ is a hereditarily Baire subspace of $\mathbb R^2$. We also show that for a subspace $E\subseteq\{(x,-x):x\in\mathbb R\}$ of the Sorgenfrey plane $\mathbb S^2$ the following conditions are equivalent: (i) $E$ is $C$-embedded in $\mathbb S^2$; (ii) $E$ is $C^*$-embedded in $\mathbb S^2$; (iii) $E$ is a countable $G_δ$-subspace of $\mathbb R^2$ and (iv) $E$ is a countable functionally closed subspace of $\mathbb S^2$.
We investigate strongly separately continuous functions on a product of topological spaces and prove that if $X$ is a countable product of real lines, then there exists a strongly separately continuous function $f:X\to\mathbb R$ which is not Baire measurable. We show that if $X$ is a product of normed spaces $X_n$, $a\in X$ and $σ(a)=\{x\in X:|\{n\in\mathbb N: x_n\ne a_n\}|<\aleph_0\}$ is a subspace of $X$, equipped with the Tychonoff topology, then for any open set $G\subseteq σ(a)$ there is a strongly separately continuous function $f:σ(a)\to \mathbb R$ such that the discontinuity point set of $f$ is equal to~$G$.
We introduce perfect cones over topological spaces and study their connection with absolute $B_1$-retracts
We present $σ$-strongly functionally discrete mappings which expand the class of $σ$-discrete mappings and generalize Banach's theorem on analytically representable functions
We study properties of strongly separately continuous mappings defined on subsets of products of topological spaces equipped with the topology of pointwise convergence. In particular, we give a necessary and sufficient condition for a strongly separately continuous mapping to be continuous on a product of an arbitrary family of topological spaces. Moreover, we charac\-terize the discontinuity point set of strongly separately continuous function defined on a subset of countable product of finite-dimensional normed spaces.
We prove that every continuous function $f:E\to Y$ depends on countably many coordinates, if $E$ is an $(\aleph_1,\aleph_0)$-invariant pseudo-$\aleph_1$-compact subspace of a product of topological spaces and $Y$ is a space with a regular $G_δ$-diagonal. Using this fact for any $α<ω_1$ we construct an $(α+1)$-embedded subspace of a completely regular space which is not $α$-embedded.
A function $f:X\to Y$ between topological spaces is said to be a {\it weakly Gibson function} if $f(\overline{G})\subseteq \overline{f(G)}$ for any open connected set \mbox{$G\subseteq X$}. We call a function $f:X\to Y$ {\it segmentary connected} if $X$ is topological vector space and $f([a,b])$ is connected for every segment $[a,b]\subseteq X$. We show that if $X$ is a hereditarily Baire space, $Y$ is a metric space, \mbox{$f:X\to Y$} is a Baire-one function and one of the following conditions holds: (i) $X$ is a connected and locally connected space and $f$ is a weakly Gibson function, (ii) $X$ is an arcwise connected space and $f$ is a Darboux function, (iii) $X$ is a topological vector space and $f$ is a segmentary connected function, then $f$ has a connected graph.
A function $f:X\to Y$ between topological spaces is said to be a {\it weakly Gibson function} if $f(\overline{U})\subseteq \overline{f(U)}$ for any open connected set \mbox{$U\subseteq X$}. We prove that if $X$ is a locally connected hereditarily Baire space and $Y$ is a $T_1$-space then an $F_σ$-measurable mapping $f:X\to Y$ is weakly Gibson if and only if for any connected set $C\subseteq X$ with the dense connected interior the image $f(C)$ is connected. Moreover, we show that each weakly Gibson $F_σ$-measurable mapping $f:\mathbb R^n\to Y$, where $Y$ is a $T_1$-space, has a connected graph.
We introduce and study $α$-embedded sets and apply them to generalize the Kuratowski Extension Theorem.
We prove that every vertically nearly separately continuous function defined on a product of a strong PP-space and a topological space and with values in a strongly $σ$-metrizable space with a special stratification, is a pointwise limit of continuous functions.
We investigate the Baire classification of mappings $f:X\times Y\to Z$, where $X$ belongs to a wide class of spaces, which includes all metrizable spaces, $Y$ is a topological space, $Z$ is an equiconnected space, which are continuous in the first variable and for a dense set in $X$ these mappings are functions of a Baire class $α$ in the second variable.
We prove that for a topological space $X$, an equiconnected space $Z$ and a Baire-one mapping $g:X\to Z$ there exists a separately continuous mapping $f:X^2\to Z$ with the diagonal $g$, i.e. $g(x)=f(x,x)$ for every $x\in X$. Under a mild assumptions on $X$ and $Z$ we obtain that diagonals of separately continuous mappings $f:X^2\to Z$ are exactly Baire-one functions, and diagonals of mappings $f:X^2\to Z$ which are continuous on the first variable and Lipschitz (differentiable) on the second one, are exactly the functions of stable first Baire class.
We introduce the notion of $B_1$-retract and investigate the connection between $B_1$- and $H_1$-retracts.
We prove that any continuous mapping $f:E\to Y$ on a completely metrizable subspace $E$ of a perfect paracompact space $X$ can be extended to a Lebesgue class one mapping $g:X\to Y$ (i.e. for every open set $V$ in $Y$ the preimage $g^{-1}(V)$ is an $F_σ$-set in $X$) with values in an arbitrary topological space $Y$.