Compact subspaces of the space of separately continuous functions with the cross-uniform topology

Abstract

We consider two natural topologies on the space $S(X\times Y,Z)$ of all separately continuous functions defined on the product of two topological spaces $X$ and $Y$ and ranged into a topological or metric space $X$. These topologies are the cross-open topology and the cross-uniform topology. We show that these topologies coincides if $X$ and $Y$ are pseudocompacts and $Z$ is a metric space. We prove that a compact space $K$ embeds into $S(X\times Y,Z)$ for infinite compacts $X$, $Y$ and a metrizable space $Z\supseteq \mathbb{R}$ if and only if the weight of $K$ is less than the sharp cellularity of both spaces $X$ and $Y$.