Hausdorff graph topology, proximal graph topology and the uniform topology for densely continuous forms and minimal USCO maps

摘要

For metric spaces (X, d x) and (Y, d y) we consider the Hausdorff metric topology $ tau _{H_rho } $ on the set (CL(X × Y), ρ) of closed subsets of the product metrized by the product (box) metric ρ and consider the proximal topology $ tau _{delta _rho } $ defined on CL(X × Y). These topologies are inherited by the set G(X, Y) of closed-graph multifunctions from X to Y, if we identify each multifunction with its graph. Finally, we consider the topology of uniform convergence τ uc on the set F(X, 2Y) of all closed-valued multifunctions, i.e. functions from X to the set (CL(Y), $ H_{d_y } $ ) of closed subsets of Y metrized by the Hausdorff metric $ H_{d_y } $ . We show the relationship between these topologies on the space G(X, Y) and also on the subspaces of minimal USCO maps and locally bounded densely continuous forms.