Hereditary normality-type properties of hyperspaces

Let X be a Hausdorff topological space and exp ( X ) be the space of all (nonempty) closed subsets of a space X with the Vietoris topology. We consider hereditary normality-type properties of exp ( X ) . In particular, we prove that if exp ( X ) is hereditarily D-normal, then X is a metrizable compact space.