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. In this 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. (C) 2012 Elsevier B.V. All rights reserved.
continuity properties alexandroff theorem vietoris topology regularity hausdorff topology monotone set multifunction