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 Z. 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 $Z\supseteq R$ if and only if the weight of K is less than the sharp cellularity of both spaces X and Y.