Some results on separate and joint continuity

Abstract

Let $f:X\times K\to \mathbb{R}$ be a separately continuous function and $\mathcal{C}$ a countable collection of subsets of $K$. Following a result of Calbrix and Troallic, there is a residual set of points $x\in X$ such that $f$ is jointly continuous at each point of $\{x\}\times Q$, where $Q$ is the set of $y\in K$ for which the collection $\mathcal{C}$ includes a basis of neighborhoods in $K$. The particular case when the factor $K$ is second countable was recently extended by Moors and Kenderov to any Čech-complete Lindelöf space $K$ and Lindelöf $\alpha$-favorable $X$, improving a generalization of Namioka's theorem obtained by Talagrand. Moors proved the same result when $K$ is a Lindelöf $p$-space and $X$ is conditionally $\sigma$-$\alpha$-favorable space. Here we add new results of this sort when the factor $X$ is ${\sigma }_{C(X)}$-$\beta$-defavorable and when the assumption "base of neighborhoods" in Calbrix-Troallic's result is replaced by a type of countable completeness. The paper also provides further information about the class of Namioka spaces.

arXiv commentTo appear in Topology and its Applications