Topological and Pointwise Upper Kuratowski Limits of a Sequence of Lower Quasi-Continuous Multifunctions

Abstract

In this  paper we deal with a connection between an upper
Kuratowski limit of a sequence of graphs of multifunctions and an
upper Kuratowski limit of a sequence of their values. Namely, we will study under which conditions for a graph (topological) cluster point $[x,y]\in X\times Y$ of a sequence $\{Gr\, F_{n}: n\in\omega\}$  of graphs of lower quasi continuous multifunctions, $y$ is a vertical (pointwise) cluster point  of the sequence $\{F_n(x): n\in\omega\}$  of values of given multifunctions. An existence of a   selection being quasi continuous on a dense open set (a dense $G_\delta$-set) for a topological (pointwise) upper Kuratowski limit is given.