The Namioka property of $KC$-functions and Kempisty spaces

A topological space $Y$ is called a Kempisty space if for any Baire space $X$ every function $f:X\times Y\to \mathbb{R}$, which is quasi-continuous in the first variable and continuous in the second variable has the Namioka property. Properties of compact Kempisty spaces are studied in this paper. In particular, it is shown that any Valdivia compact is a Kempisty space and the cartesian product of an arbitrary family of compact Kempisty spaces is a Kempisty space.