Points of continuity of semicontinuous set-valued mappings (Q1092444)

Let A(Y)(C(Y),J(Y)) be a set of all (compact, finite) nonempty subsets of a topological space Y provided with the finite, i.e. Vietoris topology. The author obtains a number of criteria for the existence of a \(G_{\delta}\)-set of continuity points of semicontinuous multivalued mappings defined on a Baire space, ranging in the above mentioned exponential spaces. A space Y has a \(G_{\delta}\)-diagonal \((G^*_{\delta}\)-diagonal; it is refined) if for any point \(y\in Y\) there exists a sequence \(\{\gamma_ n;n\in \omega \}\) of open coverings of Y such that \(\cap_{n}St(y,\gamma_ n)=\{y\}\) \((\cap_{n}\overline{St(y,\gamma_ n)}=\{y\}\); if any open neighborhood of y contains some star \(St(y,\gamma_ n))\). Examples of theorems including generalizations of two early results of P. Kenderov: If X is a Baire space, Y has a \(G^*_{\delta}\)-diagonal, F:X\(\to C(Y)\) is upper semicontinuous or Y is regular with \(G_{\delta}\)-diagonal and F:X\(\to J(Y)\) is upper semicontinuous, or Y is a refined Hausdorff space and \(F: X\to J(Y)\) is lower semicontinuous, then F has a dense \(G_{\delta}\)-set of continuity points.