Baire and weakly Namioka spaces

TL;DR

This paper explores the relationship between Baire and weakly Namioka spaces, establishing conditions under which they coincide in certain classes of topological spaces.

Contribution

It introduces the concept of weakly Namioka spaces and proves their equivalence to Baire spaces within specific classes of topological spaces.

Findings

Weakly Namioka spaces are characterized by a modified Namioka condition with second countability.

In completely regular separable and perfectly normal spaces, Baire and weakly Namioka spaces are equivalent.

Abstract

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.