Cliquishness and Quasicontinuity of Two Variables Maps

Abstract

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.