Title: Hyperspaces with a countable character of closed subsets
Abstract: For a regular space X, the hyperspace (CL(X),τF) (resp., (CL(X),τV)) is the space of all nonempty closed subsets of X with the Fell topology (resp., Vietoris topology). In this paper, we give the characterization of the space X such that the hyperspace (CL(X),τF) (resp., (CL(X),τV)) with a countable character of closed subsets. We mainly prove that (CL(X),τF) has a countable character on each closed subset if and only if X is compact metrizable, and (CL(X),τF) has a countable character on each compact subset if and only if X is locally compact and separable metrizable. Moreover, we prove that (K(X),τV) have the compact-Gδ-property if and only if X have the compact-Gδ-property and every compact subset of X is metrizable.