Lower Quasicontinuity, Joint Continuity and Related concepts

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.

💡 Research Summary

The paper investigates the joint continuity of a two‑variable mapping $f\colon X\times Y\to Z$ under very mild topological assumptions. Here $X$ and $Y$ are arbitrary topological spaces, $Z$ is a metric space, and the key hypothesis is that $Y$ possesses a countable base $\mathcal B={B_n:n\in\mathbb N}$. For each basic open set $B\in\mathcal B$ the authors consider the set‑valued map
\