Some results on separate and joint continuity

TL;DR

This paper explores conditions under which separately continuous functions are jointly continuous on large residual sets, extending previous results to broader classes of spaces and introducing new types of favorable conditions.

Contribution

It introduces new results on joint continuity for functions on product spaces, expanding the classes of spaces where Namioka-type theorems hold.

Findings

Residual sets of joint continuity points are characterized for broader space classes.

Extended results to ch-complete Lindelf3f spaces and ch-complete Lindelf3f b5-spaces.

New conditions involving ch-complete Lindelf3f spaces and countable completeness are established.

Abstract

Let $f: X\times K\to \mathbb R$ be a separately continuous function and $\mathcal C$ a countable collection of subsets of $K$. Following a result of Calbrix and Troallic, there is a residual set of points $x\in X$ such that $f$ is jointly continuous at each point of $\{x\}\times Q$, where $Q$ is the set of $y\in K$ for which the collection $\mathcal C$ includes a basis of neighborhoods in $K$. The particular case when the factor $K$ is second countable was recently extended by Moors and Kenderov to any \v{C}ech-complete Lindel\"of space $K$ and Lindel\"of $\alpha$-favorable $X$, improving a generalization of Namioka's theorem obtained by Talagrand. Moors proved the same result when $K$ is a Lindel\"of $p$-space and $X$ is conditionally $\sigma$-$\alpha$-favorable space. Here we add new results of this sort when the factor $X$ is $\sigma_{C(X)}$-$\beta$-defavorable and when the assumption "base of neighborhoods" in Calbrix-Troallic's result is replaced by a type of countable completeness. The paper also provides further information about the class of Namioka spaces.