Baire and weakly Namioka spaces

Recall that a Hausdorff space X is said to be Namioka if for every compact (Hausdorff) space Y and every metric space Z, every separately continuous function f:X×Y→Z is continuous on D×Y for some dense Gδ subset D of X. It is well known that in the class of all metrizable spaces, Namioka and Baire spaces coincide (Saint-Raymond, 1983) [23]. Further it is known that every completely regular Namioka space is Baire and that every separable Baire space is Namioka (Saint-Raymond, 1983) [23].In our paper we study spaces X, we call them weakly Namioka, for which the conclusion of the theorem for Namioka spaces holds provided that the assumption of compactness of Y is replaced by second countability of Y. We will prove that in the class of all completely regular separable spaces and in the class of all perfectly normal spaces, X is Baire if and only if it is weakly Namioka.