Lower Quasicontinuity, Joint Continuity and Related concepts

TL;DR

This paper investigates conditions under which a function of two variables is jointly continuous, quasicontinuous, or cliquish, especially when one variable space has a countable base, unifying and extending existing results.

Contribution

It establishes new conditions for joint continuity, quasicontinuity, and cliquishness of functions with respect to topological spaces with a countable base, unifying and improving prior results.

Findings

Residual set where joint continuity holds

Equivalence of joint continuity and continuity of slices

Extensions to quasicontinuity and cliquishness

Abstract

Let $X$ and $Y$ be topological spaces, let $Z$ be a metric space, and let $f: X\times Y\to Z$ be a mapping. It is shown that when $Y$ has a countable base $\mathcal B$, then under a rather general condition on the set-valued mappings $X\ni x\to f_x(B)\in 2^Z$, $B\in\mathcal B$, there is a residual set $R\subset X$ such that for every $(a,b)\in R\times Y$, $f$ is jointly continuous at $(a,b)$ if (and only if) $f_a: Y\to Z$ is continuous at $b$. Several new results are also established when the notion of continuity is replaced by that of quasicontinuity or by that of cliquishness. Our approach allows us to unify and improve various results from the literature.