Abstract: A space X is called “generalized Namioka” if for every compact space Y and every separately continuous function f : X × Y → R, there exists at least one point x ∈ X such that f is jointly continuous at each point of {x} × Y. In this paper we principally prove the following:
1. If X satisfies one of the conditions:
(1) X is a separable space of the second category;
(2) X is a pseudo-metric space of the second category;
(3) X is a countably tight space that possesses a rich family of subsets of the second category;
then X is a generalized Namioka space.
2. If X is a space of the second category and Y a W-space having a rich family of subspaces of the second category, then X × Y is of the second category.
