关于分离连续性和联合连续性的一些结果
Some results on separate and joint continuity
摘要：
设$f: X\times K\to \mathbb R$是一个独立连续的函数，而$\mathcal C$是$K$的子集的可数集合。在Calbrix和Troallic的结果之后，存在点的剩余集合$x\in X$，使得$f$在$\{x\}\times Q$的每个点处联合连续，其中$Q$是$y\in K$的集合，对于该集合，集合$\mathcal C$包括$K$中的邻域的基础。Moors和Kenderov最近将因子$K$是第二可数的特殊情况扩展到任何Čech完备的Lindelöf空间$K$和Lindelöf $α$ -有利的$X$，改进了Talagrand得到的Namioka定理的推广。Moors证明了当$K$是Lindelöf $p$ -空间且$X$是条件$σ$ - $α$ -有利空间时的相同结果。在这里，我们添加新的结果时，这种因素$X$是$σ_{C(X)}$ - $β$ -defavorable和当假设“基地的邻里”在Calbrix-Troallic的结果被替换为一种类型的可数完备性。本文还提供了有关Namioka空间类的进一步信息。
英文摘要：
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 Čech-complete Lindelöf space $K$ and Lindelöf $α$-favorable $X$, improving a generalization of Namioka's theorem obtained by Talagrand. Moors proved the same result when $K$ is a Lindelöf $p$-space and $X$ is conditionally $σ$-$α$-favorable space. Here we add new results of this sort when the factor $X$ is $σ_{C(X)}$-$β$-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.
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