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Cliquishness and Quasicontinuity of Two-Variable Maps

  Published:2011-07-08
 Printed: Mar 2013
  • A. Bouziad,
    Département de Mathématiques, Université de Rouen, UMR CNRS 6085, Saint-Étienne-du-Rouvray, France
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Abstract

We study the existence of continuity points for mappings $f\colon X\times Y\to Z$ 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 $y_n\in D_n$; 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 and item (ii) indicates that the problem of Talagrand on separately continuous maps has a positive answer for a wide class of ``small'' compact spaces.
Keywords: cliquishness, fragmentability, joint continuity, point-picking game, quasicontinuity, separate continuity, two variable maps cliquishness, fragmentability, joint continuity, point-picking game, quasicontinuity, separate continuity, two variable maps
MSC Classifications: 54C05, 54C08, 54B10, 91A05 show english descriptions Continuous maps
Weak and generalized continuity
Product spaces
2-person games
54C05 - Continuous maps
54C08 - Weak and generalized continuity
54B10 - Product spaces
91A05 - 2-person games
 

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