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) of all minimal upper semi-continuous compact-valued maps from a topological space X into a topological group Y. It is shown that, under suitable assumptions on the spaces X and Y, we may equip the set 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) 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). (C) 2013 Elsevier Ltd. All rights reserved.