下拟连续、联合连续及相关概念
Lower Quasicontinuity, Joint Continuity and Related concepts
摘要：
设$X$和$Y$是拓扑空间，$Z$是度量空间，$f: X\times Y\to Z$是映射。证明了当$Y$有可数基$\mathcal B$时，在集值映射$X\ni x\to f_x(B)\in 2^Z$，$B\in\mathcal B$的一般条件下，存在一个剩余集$R\subset X$，使得对每个$(a,b)\in R\times Y$，$f$在$(a,b)$上联合连续当且仅当$f_a: Y\to Z$在$b$上连续.当连续性的概念被拟连续性或团性的概念所取代时，也建立了几个新的结果。我们的方法使我们能够统一和改进各种文献的结果。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
