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Cliquishness and Quasicontinuity of Two-Variable Maps

  • A. Bouziad (a1)
Abstract

We study the existence of continuity points for mappings whose x-sections are fragmentable and y-sections are quasicontinuous, where X is a Baire space and Z is a metric space. For the factor Y, we consider two infinite “pointpicking” games G1(y) and G2(y) defined respectively for each y ∈ Y as follows: in the n-th inning, Player I gives a dense set Dn ⊂ Y, respectively, a dense open set Dn ⊂ Y. Then Player II picks a point yn ∈ Dn; II wins if y is in the closure of {yn : n ∈ N}, otherwise I wins. It is shown that (i) f is cliquish if II has a winning strategy in G1(y) for every y ∈ Y, and (ii) f is quasicontinuous if the x-sections of f are continuous and the set of y ∈ Y such that II has a winning strategy in G2(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.

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References
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