Characterize compact embeddings into Scro(X×Y,Z) and Scru(X×Y,Z) for general spaces

Characterize all compact spaces K that homeomorphically embed into the space S(X×Y,Z) of separately continuous functions with the cross-open topology Scro(X×Y,Z) for arbitrary topological spaces X and Y and a topological space Z, and characterize all compact spaces K that homeomorphically embed into the space S(X×Y,Z) with the cross-uniform topology Scru(X×Y,Z) when Z is a metric space.

Background

Earlier in the paper the authors fully characterize compact subspaces that embed into S(X×Y,Z) under the assumptions that X and Y are infinite compacts and Z is a metrizable space containing a copy of R (Theorem 7.1). They also show conditions under which the cross-open and cross-uniform topologies coincide (Proposition 2.1).

This problem asks for the corresponding characterization in the general setting where X and Y are not restricted to compacts and Z is only assumed to be topological (for Scro) or metric (for Scru). It broadens the scope beyond the compact case that underpins the main results of the paper.