二元映射的团性与拟连续性
Cliquishness and Quasicontinuity of Two Variables Maps
摘要：
本文研究了$x$ -截面$Y\ni y\to f(x,y)\in Z$是可分的，$y$-截面$X\ni x\to f(x,y)\in Z$是拟连续的映射$f: X\times Y\to Z$的连续点的存在性，其中$X$是Baire空间，$Z$是度量空间.对于因子$Y$，我们考虑两个无限的“选点”游戏$G_1(y)$和$G_2(y)$，分别为每个$y\in Y$定义如下：在$n$第一局，参与者I分别给出一个稠密集$D_n\subset Y$，一个稠密开集$D_n\subset Y$，然后参与者II选择一个点$y_n\inD_n$ ;如果$y$在$\{y_n:n\in\mathbb N\}$的闭包中，II获胜，否则I获胜。证明了：（i）$f$是团的，如果对每个$y\in Y$，II在$G_1(y)$中有获胜策略;（ii）$f$是拟连续的，如果$f$的$x$ -部分是连续的，且使得II在$G_2(y)$中有获胜策略的$y\in Y$的集合在$Y$中是稠密的.第（i）项实质上扩展了Debs（1986）的结果，第（ii）项表明Talagrand（1985）关于分离连续映射的问题对广泛的“小”紧空间有肯定的答案。
英文摘要：
We study the existence of continuity points for mappings $f: X\times Y\to Z$ whose $x$-sections $Y\ni y\to f(x,y)\in Z$ are fragmentable and $y$-sections $X\ni x\to f(x,y)\in Z$ are quasicontinuous, where $X$ is a Baire space and $Z$ is a metric space. For the factor $Y$, we consider two infinite "point-picking" games $G_1(y)$ and $G_2(y)$ defined respectively for each $y\in Y$ as follows: In the $n$th inning, Player I gives a dense set $D_n\subset Y$, respectively, a dense open set $D_n\subset Y$, then Player II picks a point $y_n\inD_n$; II wins if $y$ is in the closure of $\{y_n:n\in\mathbb N\}$, otherwise I wins. It is shown that (i) $f$ is cliquish if II has a winning strategy in $G_1(y)$ for every $y\in Y$, and (ii) $f$ is quasicontinuous if the $x$-sections of $f$ are continuous and the set of $y\in Y$ such that II has a winning strategy in $G_2(y)$ is dense in $Y$. Item (i) extends substantially a result of Debs (1986) and item (ii) indicates that the problem of Talagrand (1985) on separately continuous maps has a positive answer for a wide class of "small" compact spaces.
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