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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 continuousfunction $f:X\times{Y}\rightarrow{Z}$ is continuous on $D\times{Y}$ for somedense $G_\delta$ subset $D$ of $X$. It is well known that in the class of allmetrizable spaces, Namioka and Baire spaces coincide (Saint-Raymond, 1983).Further it is known that every completely regular Namioka space is Baire andthat every separable Baire space is Namioka (Saint-Raymond, 1983).  In our paper we study spaces $X$, we call them weakly Namioka, for which theconclusion of the theorem for Namioka spaces holds provided that the assumptionof compactness of $Y$ is replaced by second countability of $Y$. We will provethat in the class of all completely regular separable spaces and in the classof all perfectly normal spaces, $X$ is Baire if and only if it is weaklyNamioka.

Baire and weakly Namioka spaces