An ascoli theorem for multi-valued functions

Abstract

The concept of simultaneous or collective continuity of a family of single valued functions was introduced by Gale [3] for regular spaces to replace equicontinuiry in metric spaces. Smithson [6] extended the standard point-open and compact-open function space topologies to include multi-valued functions. The aim of this paper is to use these topologies and extend the notion of collective continuity in order to obtain an Ascoli type theorem for multi-valued functions analogous to Theorem 1 in [3, p. 304]. We have the following theorem in mind: