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

We consider two natural topologies on the space S(X x 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 x Y, Z) for infinite compacts X, Y and a metrizable space Z superset of Rif and only if the weight of K is less than the sharp cellularity of both spaces X and Y. (c) 2024 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC license (http://creativecommons .org /licenses /by -nc /4 .0/).