On the regularity of the topology of separate continuity

Given topological spaces X,Y, there is a unique topology T+ on X×Y such that, for all topological spaces Z, a function f :X×Y→Z is continuous with respect to T+ iff f is separately continuous. We consider situations under which T+ is regular or normal. This is related to Eberlein compacta in the case that X,Y are compact, and to σ-sets in the case that X,Y are separable metric.