Baire classification of separately continuous functions and Namioka property

Abstract

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).