Lebesgue measurability of separately continuous functions and separability

TL;DR

This paper explores the relationship between separability and the countable chain condition in spaces with the $L$-property, showing that certain conditions imply separability and providing a counterexample to a previous question.

Contribution

It establishes that completely regular Baire spaces with the $L$-property and countable chain condition are separable, and constructs a nonseparable example, answering Burke's question negatively.

Findings

Every completely regular Baire space with the $L$-property and countable chain condition is separable.

A nonseparable completely regular space with the $L$-property and countable chain condition exists.

Abstract

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.