On the Choquet]Dolecki Theorem

In this paper, we prove that if a multifunction F: TªX from a first countable space T into a space X with property ). is upper semicontinuous at a point t0gT, then the active boundary of F at t0is compact. Moreover, we also show that if X is an angelic space, then the active boundary of F at t0 is the smallest kernel of F at t0.