Joint continuity of separately continuous functions

Abstract:

It is shown that a separately continuous function $f:X \times Y \to Z$ from the product of a certain type of Hausdorff space $X$ and a compact Hausdorff space $Y$ into a metrizable space $Z$ is jointly continuous on a set of the type $A \times Y$, where $A$ is a dense ${G_\delta }$ set in $X$. The class of Hausdorff spaces $X$ in question is defined by a gametheoretic condition. The result improves (and simplifies the proof of) a recent result of Namioka. Many "deep" theorems in functional analysis and automatic continuity theory are easy corollaries.