Abstract
For a space X, let (CL(X),τV), (CL(X),τlocfin) and (CL(X),τF) be the set CL(X) of all nonempty closed subsets of X which are endowed with Vietoris topology, locally finite topology and Fell topology respectively. We prove that (CL(X),τV) is quasi-metrizable if and only if X is a separable metrizable space and the set of all non-isolated points of X is compact, (CL(X),τlocfin) is quasi-metrizable or symmetrizable if and only if X is metrizable and the set of all non-isolated points of X is compact, and (CL(X),τF) is quasi-metrizable if and only if X is hemicompact and metrizable. As an application, we give a negative answer to a Conjecture in [13].
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