Topologies for $2^{x}$; set-valued functions and their graphs

Abstract:

We consider the problem of topologizing ${2^X}$, the set of all closed subsets of a topological space $X$, in such a way as to make continuous functions from a space $Y$ into ${2^X}$ precisely those functions with closed graphs. We show there is at most one topology with this property, and if $X$ is a regular space, the existence of such a topology implies that $X$ is locally compact. We then define the compact-open topology for ${2^X}$, which has the desired property for locally compact Hausdorff $X$. The space ${2^X}$ with this topology is shown to be homeomorphic to a space of continuous functions with the well-known compact-open topology. Finally, some additional properties of this topology are discussed.