On lower semicontinuous multifunctions in quasi-uniform and vector spaces (Q2782229)

Let \(X\) be a topological space and \(Y\) a quasi-uniform space endowed with a cover \({\mathcal B}\). A notion of \({\mathcal B}\)-lower semicontinuity of a multifunction \(F:X\multimap Y\) is introduced and investigated. Special cases of Hausdorff and Vietoris lsc correspond to \({\mathcal B}=\{Y\}\) and \({\mathcal B}=\{\{y\}: y\in Y\}\) resp. Conditions under which \({\mathcal B}\)-lower semicontinuity is preserved with respect to creating unions, products, intersections, linear sums and convex hull in corresponding structures of space \(Y\) are exhibited. A simple counterexample showing that the \(\text{conv} F\) may fail to be \(H\)-lsc for \(F\) being \(H\)-lsc, but \(Y\) not a locally convex linear topological space, is also given.NEWLINENEWLINEFor the entire collection see [Zbl 0983.00046].