Onepoint discontinuity set of separately continuous functions on the product of two compact spaces

It is investigated the existence of a separately continuous function $f:X\times Y\to \mathbb R$ with an onepoint set of discontinuity for topological spaces $X$ and $Y$ which satisfy compactness type conditions. In particular, it is shown that for compact spaces $X$ and $Y$ and nonizolated points $x_0\in X$ and $y_0\in Y$ there exists a separately continuous function $f:X\times Y\to \mathbb R$ with the set $\{(x_0,y_0)\}$ of discontinuity points if and only if there exist sequences of nonempty functional open sets which converge to $x_0$ and $y_0$ in $X$ and $Y$ respectively

Construction of separately continuous functions of nn variables with given restriction

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 $x\in X$

On questions which are connected with Talagrand problem

We prove the following results. 1. If $X$ is a $\alpha$-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 $\beta\mathbb N$. 2. There exist a $\alpha$-favourable space $X$, a dense in $\beta\mathbb N\setminus\mathbb N$ 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 $C_p(\beta\mathbb N\setminus\mathbb N,\{0,1\})$ of all continuous functions $x:\beta\mathbb N\setminus\mathbb N\to\{0,1\}$ with the topology of point-wise convergence is a Baire space

Lebesgue measurability of separately continuous functions and separability

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 $F_{\sigma}$-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

The discontinuity points set of separately continuous functions on the products of compacts

It is solved a problem of construction of separately continuous functions on the product of compacts with a given discontinuity points set. We obtaine the following results. 1. For arbitrary \v{C}ech complete spaces $X$, $Y$ and a separable compact perfect projectively nowhere dense zero set $E\subseteq X\times Y$ there exists a separately continuous function $f:X\times Y\to\mathbb R$ the discontinuity points set of which equals to $E$. 2. For arbitrary \v{C}ech complete spaces $X$, $Y$ and nowhere dense zero sets $A\subseteq X$ and $B\subseteq Y$ there exists a separately continuous function $f:X\times Y\to\mathbb R$ such that the projections of the discontinuity points set of $f$ coincide with $A$ and $B$ respectively. An example of Eberlein compacts $X$, $Y$ and nowhere dense zero sets $A\subseteq X$ and $B\subseteq Y$ such that the discontinuity points set of every separately continuous function $f:X\times Y\to\mathbb R$ does not coincide with $A\times B$, and $CH$-example of separable Valdivia compacts $X$, $Y$ and separable nowhere dense zero sets $A\subseteq X$ and $B\subseteq Y$ such that the discontinuity points set of every separately continuous function $f:X\times Y\to\mathbb R$ does not coincide with $A\times B$ are constructed

Continuous extension of functions from countable sets

We give a characterization of countable discrete subspace $A$ of a topological space $X$ such that there exists a (linear) continuous mapping $\varphi:C_p^*(A)\to C_p(X)$ with $\varphi(y)|_A=y$ for every $y\in C_p^*(A)$. Using this characterization we answer two questions of A.~Arhangel'skii. Moreover, we introduce the notion of well-covered subset of a topological space and prove that for well-covered functionally closed subset $A$ of a topological space $X$ there exists a linear continuous mapping $\varphi:C_p(A)\to C_p(X)$ with $\varphi(y)|_A=y$ for every $y\in C_p(A)$

Separate continuity topology and a generalization of Sierpinski's theorem

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

Baire classification of separately continuous functions and Namioka property

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 Lindel\"of 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 $X$ $G_{\delta}$-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)

The Namioka property of KCKC-functions and Kempisty spaces

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

Namioka spaces and strongly Baire spaces

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 $\beta-\sigma$-unfavorable space is a strongly Baire space