Linear and quasi-linear spaces of set-valued maps

In this paper we investigate the algebraic structure of certain spaces of set-valued maps. Among other results, we show that for an arbitrary topological space XX and a metrisable topological vector space ZZ, the space M(X,Z)M(X,Z) of minimal upper semi-continuous compact valued (musco) maps from XX into ZZ is a linear space. This result extends a previously known result on the linear structure of spaces of musco maps. Previously, this result was known only in the case when XX is a Baire space. We also study topologies of uniform convergence on compact sets on M(X,Z)M(X,Z).