Quasi-continuous and cliquish selections of multifunctions on product spaces (Q1803975)

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 \textit{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.