Cliquishness and Quasicontinuity of Two Variables Maps

TL;DR

This paper investigates the continuity points of functions of two variables with fragmentable and quasicontinuous sections, using infinite point-picking games to characterize cliquishness and quasicontinuity in Baire and metric spaces.

Contribution

It introduces game-theoretic conditions for cliquishness and quasicontinuity of two-variable maps, extending previous results and addressing Talagrand's problem.

Findings

Characterization of cliquishness via winning strategies in point-picking games.

Conditions for quasicontinuity based on dense sets of points with winning strategies.

Extension of Debs's results and positive resolution of Talagrand's problem for certain spaces.

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.