Separation properties in X and \(2^ X\). Upper semicontinuous and measurable multifunctions
There are close connections between separation properties of a topological space X and its hyperspace \(2^ X\), and among the semicontinuity properties of a relation F from a topological space T to X, continuity of F regarding it as a function from T to \(2^ X\) (or in general to \(\wp (X)\), equipped with suitable topologies) and closedness of the graph \(Gr(F)\) or \(Gr(\bar F)\). In problems of measurable selections of F, one takes T to be a suitable measurable space. The author improves and/or sharpens, in both these contexts, some known results detailing the suggested connections using the properties of weak regularity of X, countable and weak countable separation in X and \(2^ X\), and closedness of graphs \((Gr(F,g)\), \(\Omega\) etc. For instance it is proved that weak regularity of X is equivalent to the property of having \(Gr(F)\) closed for every continuous (or u.s.c.) F; and that X is countably R-separated iff X is an \(R_ 0\)-space and \((2^ X,\sigma ({\mathcal V}^+))\) is countably separated.
scientific article; zbMATH DE number 4113512