Kuratowski convergence on compact sets

Let X be a metric space, and C the collection of all nonempty closed subsets of X. Let G = {graph(f,Ω): Ω ∈ C and f:X → Rm is continuous}, where graph(f,Ω) = {(x, f (x)) ∈ X × Rm : x ∈ Ω}. The notion of cK-convergence (Kuratowski convergence on compact sets) in G is defined and characterized, and its relationships with other types of convergence in G are explored. These results allow a comparison between the topology induced by cK-convergence and various other graph space topologies.