In addition, several theorems on the first countability of the space of compact-valued functions with one of the compact-open topologies of Smithson are proved. One such theorem is the following generalization of a result of R. Arens: If X is a Hausdorff hemicompact space and Y is a Hausdorff locally compact second countable space, then the collection of upper semicontinuous compact-valued functions a from X to Y satisfying a(a–1(F) Ì F for each F Ì Y is first countable.