Minimal usco and minimal cusco maps (Q2800612)

For topological spaces \(X\) and \(Y\), a set-valued map \(F:X \rightsquigarrow Y\) is said to be usco if it is upper semicontinuous and nonempty compact-valued. When \(Y\) is a linear topological space, an usco convex-valued map \(F:X\rightsquigarrow Y\) is said to be cusco. An usco (resp. cusco) map \(F:X\rightsquigarrow Y\) is said to be minimal if for every usco (resp. cusco) map \(G:X\rightsquigarrow Y\) with \(G(x) \subset F(x)\), \(x \in X\), we have \(G=F\).NEWLINENEWLINEPresenting a survey of known results concerning minimal usco and minimal cusco maps, the authors give characterizations of minimal usco and minimal cusco maps using quasicontinuous selections and subcontinuous ones. Here, a single-valued function \(f: X\to Y\) is said to be quasicontinuous if for every \(x \in X\) and every neighborhoods \(U\) and \(V\) of \(x\) and \(f(x)\), respectively, there exits an open subset \(W\) of \(X\) such that \(W \subset U\) and \(f(W) \subset V\), and \(f: X\to Y\) is said to be subcontinuous if for every \(x \in X\) and every net \(\{x_\sigma:\sigma\in\Sigma\}\) in \(X\) converging to \(x\), there is a convergent subnet of \(\{f(x_\sigma):\sigma\in\Sigma\}\). A single-valued function \(f: X\to Y\) is called a selection of a set-valued map \(F:X\rightsquigarrow Y\) if \(f(x)\in F(x)\) for every \(x \in X\).NEWLINENEWLINEFor a topological space \(X\) and a Banach space \(Y\), let \(MU(X,Y)\) (resp. \(MC(X,Y)\)) denote the set of all usco (resp. cusco) maps from \(X\) to \(Y\). The authors also discuss a bijection between \(MU(X,Y)\) and \(MC(X,Y)\) and related function spaces.