  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 $f:X\times{Y}\rightarrow{Z}$ is continuous on $D\times{Y}$ for some
dense $G_\delta$ subset $D$ of $X$. It is well known that in the class of all
metrizable spaces, Namioka and Baire spaces coincide (Saint-Raymond, 1983).
Further it is known that every completely regular Namioka space is Baire and
that every separable Baire space is Namioka (Saint-Raymond, 1983).
  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.
