On generalized Namioka spaces and joint continuity of functions on product of spaces

Abstract

A space $X$ is called a generalized Namioka space (g$\mathcal{N}$-space), if for every compact space $Y$ and every separately continuous function $f\colon X\times Y\rightarrow\mathbb{R}$, there exists at least one point $x\in X$ such that $f$ is jointly continuous at each point of $\{x\}\times Y$. We principally prove the following results: (1) If $X=\prod_{α\in A}X_α$ is non-meager such that each factor is a separable space or each factor is a pseudo-metric space, then $X$ is a g$\mathcal{N}$-space. (2) If $X$ is a separable space and $Y$ a pseudo-metric space such that $X\times Y$ is Baire (resp. non-meager), then $X\times Y$ is an $\mathcal{N}$-space (resp. a g$\mathcal{N}$-space). (3) If $X=\prod_{α\in A}X_α$ such that each factor is separable and $\prod_{α\in A^\prime}X_α$ is a non-meager space for each countable subset $A^\prime$ of $A$, then $X$ is a non-meager g$\mathcal{N}$-space. (4) If $X=\prod_{α\in A}X_α$ such that each factor has a countable $π$-base, then each tail set having the property of Baire in $X$ is either meager or residual. If $G$ is a g$\mathcal{N}$ right-topological group and $X$ a locally compact regular space, or, if $G$ is a separable first countable non-meager right-topological group and $X\times X$ a countably compact completely regular space, then any separately continuous action $G\curvearrowright X$ is jointly continuous.