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

Abstract

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.