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 X X into a topological group Y Y . It is shown that, under suitable assumptions on the spaces X X and Y Y , 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 ) .
set-valued maps algebraic operations convergence structure interval functions