Boundaries of and selectors for upper semi-continuous multi-valued maps
A multi-valued map F from a topological space X into a topological space Y is said to be upper semi-continuous (u.s.c.) if \(\{\) x:F(x)\(\subset U\}\) is open in X for each open set U in Y. The boundary of F at a point \(x_ 0\) in X is the subset of \(F(x_ 0)\) defined by \[ K(x_ 0)=F(x_ 0)\cap \cap_{V\in {\mathcal V}}cl[F(V)\backslash F(x_ 0)], \] where \({\mathcal V}\) is the set of all neighbourhoods of \(x_ 0\). It is proved that if \(x_ 0\) is a first countable point in a Hausdorff space X and F is an u.s.c. map of X into an angelic space Y, then \(K(x_ 0)\) is compact; if \(x_ 0\) is a q-point (in the sense of Michael) in a regular Hausdorff space X, F is an u.s.c. map of X into a Dieudonné complete space Y, and each point of \(Y\backslash F(x_ 0)\) is contained in a \({\mathcal G}_{\delta}\)-set disjoint from \(F(x_ 0)\), then \(K(x_ 0)\) is compact. These compactness results are used to extend earlier measurable selection results of Jayne and Rogers. It is proved that if F is an u.s.c. map, with arbitrary non-empty values, from a metric space X into the Banach space \(c_ 0(\Gamma)\) with its weak topology, then F has a selector f of the second Borel class using the norm topology of \(c_ 0(\Gamma)\); if F is an u.s.c. map, with arbitrary non-empty values, from a metric space X into a Banach space Y with its weak topology, and if Y is K-analytic in its weak topology, then F has a Baire measurable selector, using the weak topology on Y.
scientific article; zbMATH DE number 3866162
scientific article