Cliquishness and Quasicontinuity of Two-Variable Maps

Abstract

Abstract. We study the existence of continuity points for mappings f : X × Y → Z whose x-sections Y ∋ y → f (x, y) ∈ Z are fragmentable and y-sections X ∋ x → f (x, y) ∈ Z are quasicontinuous, where X is a Baire space and Z is a metric space. For the factor Y , we consider two infinite "pointpicking" games G 1 (y) and G 2 (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 G 1 (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 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.

On a Problem of Talagrand Concerning Separately Continuous Functions

“…The mapping e : E × K → {0, 1} constructed in § § 2 and 3 gives rise to the following result, related to the works by Bouziad [2], Debs [6] and Mykhaylyuk [21]. Proposition 4.1.…”

“…The function φ constructed in the proof of Proposition 4.1 is also quasi-continuous in the second variable. We refer the reader to [2] and [21] for some positive results concerning points of joint continuity of functions of two variables, continuous in one variable and quasi-continuous in the other one.…”

“…The function e : E × K → {0, 1} we shall construct in the proof of Theorem 1.2 gives rise to a function ϕ : E × X → {0, 1}, where X is a compact space with the Namioka property, such that ϕ is continuous in the first variable, upper-semicontinuous in the second variable and has no points of joint continuity. This topic, related to the works by Bouziad [2], Debs [6] and Mykhaylyuk [21], is discussed in § 4, cf. Comment 5.4.…”

Points of Continuity of Quasi-Continuous Functions

“…A function f : X × Y → Z is called a KC-function if it is quasi-continuous with respect to the first variable and continuous with respect to the second variable. A few mathematicians studied points of joint continuity of KC-functions see, e.g., [3,6,10,16,17,20,22,23] and the survey paper of Neubrunn [25]). In particular, Piotrowski [26] proved the following result.…”