Approximation theorems for fuzzy set multifunctions in Vietoris topology.\\ Physical implications of regularity

In this paper, we consider continuity properties (especially, regularity, also viewed as an approximation property) for $% \mathcal{P}_{0}(X)$-valued set multifunctions ($X$ being a linear, topological space), in order to obtain Egoroff and Lusin type theorems for set multifunctions in the Vietoris hypertopology. Some mathematical applications are established and several physical implications of the mathematical model of regularity are presented, which allows a classification of the physical models.