Continuity properties and Alexandroff theorem in Vietoris topology

In several previous papers, we introduced and studied various continuity properties in the set-valued case. Precisely, we established results concerning increasing/decreasing convergence, exhaustivity, order continuity and regularity in Hausdorff topology for closed-set-valued monotone set multifunctions on a real normed space. s paper, taking as starting point of a previous work, we introduce and study from another point of view, corresponding continuity properties in Vietoris topology for more general monotone set multifunctions on a Hausdorff linear topological space. Thus, we obtain generalizations of some previous results, which in turn, generalize to the set-valued case, well known results from classical fuzzy (i.e., monotone) measure theory. An Alexandroff theorem in Vietoris topology and its converse are also established.