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Quasi-continuous and cliquish selections of multifunctions on product spaces. (English) Zbl 0782.54019
Let $$X$$ be a topological space and let $$M$$ be metric. Further, let $$K(M)$$ be the family of nonempty compact subsets of $$M$$, and let $$S_ \varepsilon(A)$$ denote an $$\varepsilon$$-neighborhood of $$A$$. A multifunction $$F:X \to K(M)$$ is said to be cliquish if for every $$x_ 0 \in X$$ for every $$\varepsilon>0$$ and any neighborhood $$U$$ of $$x_ 0$$ there is a nonempty open set $$V \subset U$$ such that $$\bigcap_{x \in V}S_ \varepsilon \bigl( F(x) \bigr) \neq \emptyset$$. A selection of $$F$$ is any function $$f:X \to Y$$ such that $$f(x) \in F(x)$$ for $$x \in X$$.
Sample results; (1) Let $$X$$ be Baire and $$M$$ be separable metric. A multifunction $$F:X \to K(M)$$ is cliquish if and only if it has a cliquish selection. (2) Let $$X$$ be Baire and $$Y$$ be locally of $$\pi$$-countable type. Let $$F:X\times Y\to K(M)$$. Let $$F_ x$$ be $$h_ d$$-cliquish for any $$x \in S$$, $$S$$ being of first category, and $$F_ y$$ be $$u$$-$$B$$- continuous for $$y \in Y$$. Then $$F$$ is cliquish.
The latter result is a multifunction version of a theorem of E. Wingler and the reviewer [ibid. 16, No. 2, 408-414 (1991; Zbl 0733.26006)]. The paper ends with, actually three, interesting questions pertaining to cliquishness and quasi-continuity of multifunctions.

##### MSC:
 54C65 Selections in general topology 54C60 Set-valued maps in general topology 54C08 Weak and generalized continuity
Zbl 0733.26006