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

TL;DR

This paper investigates generalized Namioka spaces, establishing conditions under which product spaces and group actions exhibit joint continuity of functions, extending classical results in topology.

Contribution

It introduces new classes of generalized Namioka spaces and proves their properties in product spaces and group actions, broadening the understanding of joint continuity.

Findings

Product spaces with certain properties are gN-spaces.

Separately continuous actions become jointly continuous under specified conditions.

Conditions for non-meager and Baire spaces ensure joint continuity.

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 $X=\prod_{\alpha\in A}X_\alpha$ 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 $X=\prod_{\alpha\in A}X_\alpha$ such that each factor is separable and $\prod_{\alpha\in A^\prime}X_\alpha$ 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 $X=\prod_{\alpha\in A}X_\alpha$ such that each factor has a countable $\pi$-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.