arXiv subjects
Publications and source records attributed to Volodymyr Mykhaylyuk.
We construct a separately continuous function $e:E\times K\to\{0,1\}$ on the product of a Baire space $E$ and a zero-dimensional compact space $K$ such that no restriction of $e$ to any non-meager Borel set in $E\times K$ is continuous. The function $e$ provides a negative solution of Talagrand's problem in \cite{T}.
We compare possibilities of extension of bounded and unbounded Baire-one functions from subspaces of topological 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
We investigate the possibility of extension of $F_σ$-measurable and Baire-one maps from subspaces of topological spaces when these maps take values in spaces which covers by a sequence of metrizable spaces with special properties
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 $x\in X$.
We introduce a functional Lebesgue classification of multivalued mappings and obtain results on upper and lower Lebesgue classifications of multivalued mappings $F:X\times Y\to Z$ for wide classes of spaces $X$, $Y$ and $Z$.
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 $βX$.
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.
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.
It is shown that for any Baire space $X$, linearly ordered compact spaces $Y_1,\dots, Y_n$, compact space $Y\subseteq Y_1\times\cdots \times Y_n$ such that for every parallelepiped $W\subseteq Y_1\times\cdots \times Y_n$ the set $Y\cap W$ is connected, and separately continuous mapping $f:X\times Y\to\mathbb R$ there exists a dense in $X$ $G_δ$-set $A\subseteq X$ such that $f$ is jointly continuous at every point of $A\times Y$.
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.
It is shown that for a function $f:\mathbb R^2\to \mathbb R$ which is measurable with respect to the first variable and upper semicontinuous quasicontinuous and increasing with respect to the second variable there exists a Caratheodory's solution $y(x)=y_0+\int\limits_{x_0}^xf(t,y(t))dμ(t)$ of the Cauchy problem $y'(x)=f(x,y(x))$ with the initial condition $y(x_0)=y_0$. There are constructed examples which indicate to essentiality of condition of increasing and give the negative answer to a question of Z.~Grande.
A negative solution of Problem 188 posed by Max Eidelheit in {\it the Scottish Book} concerning superpositions of separately absolutely continuous functions is presented. We discuss here his and some related problems which have also negative solutions. Finally, we give an explanation of such negative answers from the "embeddings of Banach spaces" point of view.
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 $x\in X$.
Our main technical tool is a principally new property of compact narrow operators which works for a domain space without an absolutely continuous norm. It is proved that for every Köthe $F$-space $X$ and for every locally convex $F$-space $Y$ the sum $T_1+T_2$ of a narrow operator $T_1:X\to Y$ and a compact narrow operator $T_2:X\to Y$ is a narrow operator. This gives a positive answers to questions asked by M.~Popov and B.~Randrianantoanina (\cite[Problem 5.6 and Problem 11.63]{PR})
We prove that a function $f(x,y)$ of real variables defined on a rectangle, having square integrable partial derivatives $f"_{xx}$ and $f"_{yy}$, has almost everywhere mixed derivatives $f"_{xy}$ and $f"_{yx}$.
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
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.