Minimal usco and minimal cusco maps and compactness

Abstract

We prove a nice generalization of the Arzela–Ascoli Theorem from continuous functions to minimal usco/cusco maps into metric spaces. Let X be a locally compact space, (Y,d) be a metric space, K(Y) be the space of all nonempty compact subsets of Y and MU(X,Y) be the space of minimal usco maps from X to Y. The family E⊆MU(X,Y) is compact in K(Y)X equipped with the topology τUC of uniform convergence on compact sets if and only if E is closed in (MU(X,Y),τUC), pointwise bounded and densely equicontinuous. The same result holds also for compact subsets of (MC(X,Y),τUC), the space of minimal cusco maps from X to a Banach space Y.