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

TL;DR

This paper constructs separately continuous functions on products of compact spaces with prescribed discontinuity sets, advancing understanding of their possible discontinuity structures and providing specific examples and counterexamples.

Contribution

It introduces methods to construct separately continuous functions with specified discontinuity sets on products of compact spaces, including new examples and counterexamples.

Findings

Existence of functions with prescribed discontinuity sets on products of Čech complete spaces.

Construction of functions with projections of discontinuity sets matching given zero sets.

Counterexamples showing limitations in representing certain zero sets as discontinuity sets.

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.