arXiv subjects
Publications and source records attributed to V. V. Mykhaylyuk.
It is solved the problem on construction of separately continuous functions on product of $n$ topological spaces with given restriction. In particular, it is shown that for every topological space $X$ and $n-1$ Baire class function $g:X\to \mathbb R$ there exists a separately continuous function $f:X^n\to\mathbb R$ such that $f(x,x,\dots,x)=g(x)$ for every $x\in X$.
It is solved the problem on constructed of separately continuous functions on product of two topological spaces with given restriction. In particular, it is shown that for every topological space $X$ and first Baire class function $g:X\to \bf R$ there exists separately continuous function $f:X\times X \to \bf R$ such that $f(x,x)=g(x)$ for every $x\in X$.
The separately continuity topology is considered and some its properties are investigated. With help of these properties a generalization of Sierpinski theorem on determination of real separately continuous function by its values on an arbitrary dense set is obtained.
It is shown that for any Baire space $X$, linearly ordered compact $Y$ 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 $A\times Y$, i.e. any linearly ordered compact is a co-Namioka space.
A notion of strongly Baire space is introduced. Its definition is a transfinite development of some equivalent reformulation of the Baire space definition. It is shown that every strongly Baire space is a Namioka space and every $β-σ$-unfavorable space is a strongly Baire space.
We prove the following two results. 1. If $X$ is a completely regular space such that for every topological space $Y$ each separately continuous function $f:X\times Y\to\mathbb R$ is of the first Baire class, then every Lindelöf subspace of $X$ bijectively continuously maps onto a separable metrizable space. 2. If $X$ is a Baire space, $Y$ is a compact space and $f:X\times Y\to\mathbb R$ is a separately continuous function which is a Baire measurable function, then there exists a dense in $X$ $G_δ$-set $A$ such that $f$ is jointly continuous at every point of $A\times Y$ (this gives a positive answer to a question of G. Vera).
We prove the following results. 1. If $X$ is a $α$-favourable space, $Y$ is a regular space, in which every separable closed set is compact, and $f:X\times Y\to\mathbb R$ is a separately continuous everywhere jointly discontinuous function, then there exists a subspace $Y_0\subseteq Y$ which is homeomorphic to $β\mathbb N$. 2. There exist a $α$-favourable space $X$, a dense in $β\mathbb N\setminus\mathbb N$ countably compact space $Y$ and a separately continuous everywhere jointly discontinuous function $f:X\times Y\to\mathbb R$. Besides, it was obtained some conditions equivalent to the fact that the space $C_p(β\mathbb N\setminus\mathbb N,\{0,1\})$ of all continuous functions $x:β\mathbb N\setminus\mathbb N\to\{0,1\}$ with the topology of point-wise convergence is a Baire space.
It is investigated necessary and sufficient conditions on topological spaces $X=\prod\limits _{s\in S}X_s$ and $Y=\prod\limits _{t\in T}Y_t$ for the dependence of every separately continuous functions $f:X\times Y\to \mathbb R$ on at most $\aleph$ coordinates with respect to the first or the second variable.
It is obtained necessary and sufficient conditions of dependence on $\aleph$ coordinates for functions of several variables, each of which is a product of metrizable factors. The set of discontinuity points of such functions is characterized in the case, when each variable is a product of separable metrizable spaces.
It is shown that a set in product of $n$ metrizable spaces is the discontinuity points set of some separately continuous function if and only if this set can be represented as the union of a sequence of $F_σ$-sets which are locally projectively meager.
It is proved that a differentiable with respect to each variable function $f:\mathbb R^2\to\mathbb R$ is a solution of the equation $ \frac{\partial u}{\partial x} + \frac{\partial u}{\partial y}=0$ if and only if there exists a function $φ:\mathbb R\to\mathbb R$ such that $f(x,y)=φ(x-y)$. This gives a positive answer to a question of R.~Baire. Besides, we use this result to solving analogous partial differential equations in abstract spaces and partial differential equations of higher-order.
We introduce a class of $β-v$-unfavorable spaces, which contains some known classes of $β$-unfavorable spaces for topological games of Choquet type. It is proved that every $β-v$-unfavorable space $X$ is a Namioka space, that is for any compact space $Y$ and any separately continuous function $f:X\times Y\to \mathbb R$ there exists a dense in $X$ $G_δ$-set $A\subseteq X$ such that $f$ is jointly continuous at each point of $A\times Y$.
It is studied a connection between the separability and the countable chain condition of spaces with the $L$-property (a topological space $X$ has the $L$-property if for every topological space $Y$, separately continuous function $f:X\times Y\to\mathbb R$ and open set $I\subseteq \mathbb R$ the set $f^{-1}(I)$ is a $F_σ$-set). We show that every completely regular Baire space with the $L$-property and the countable chain condition is separable and construct a nonseparable completely regular space with the $L$-property and the countable chain condition. This gives a negative answer to a question of M.~Burke.
A topological space $Y$ is called a Kempisty space if for any Baire space $X$ every function $f:X\times Y\to\mathbb R$, which is quasi-continuous in the first variable and continuous in the second variable has the Namioka property. Properties of compact Kempisty spaces are studied in this paper. In particular, it is shown that any Valdivia compact is a Kempisty space and the cartesian product of an arbitrary family of compact Kempisty spaces is a Kempisty space.
We prove that every point-finite family of nonempty functionally open sets in a topological space $X$ has the cardinality at most an infinite cardinal $κ$ if and only if $w(X)\leqκ$ for every Valdivia compact space $Y\subseteq C_p(X)$. Correspondingly a Valdivia compact space $Y$ has the weight at most an infinite cardinal $κ$ if and only if every point-finite family of nonempty open sets in $C_p(Y)$ has the cardinality at most $κ$, that is $p(C_p(Y))\leq κ$. Besides, it was proved that $w(Y)=p(C_p(Y))$ for every linearly ordered compact $Y$. In particular, a Valdivia compact space or linearly ordered compact space $Y$ is metrizable if and only if $p(C_p(Y))=\aleph_0$. This gives answer to a question of O.~Okunev and V.~Tkachuk.