Algebraic and topological structure of some spaces of set-valued maps

In this paper we consider possible topological and algebraic structures on some spaces of set-valued maps. In particular, we introduce algebraic operations on the set M(X,Y)M(X,Y) of all minimal upper semi-continuous compact-valued maps from a topological space XX into a topological group YY. It is shown that, under suitable assumptions on the spaces XX and YY, we may equip the set M(X,Y)M(X,Y) with a group structure. This structure extends the usual pointwise operations on the set of point-valued continuous functions. We also introduce convergence structures on certain sets of set-valued maps. In particular, we consider the continuous convergence structure on sets of upper semi-continuous maps, as well as a convergence structure on M(X,Y)M(X,Y) derived through it, which is compatible with the mentioned algebraic structure. It is also shown that the generalized compact-open topology is compatible with the algebraic structure introduced on M(X,Y)M(X,Y).