Some characterizations of weak sobriety and upper spaces

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Weak sobriety is a topological property that gives rise to a categorical equivalence between topological spaces and distributive lattices without a least element. In this paper, we give some characterizations of weak sobriety and obtain some characterizations of local compactness, core-compactness and coherence of the special topological space by its upper space, where the upper space of a space is the set of all nonempty saturated compact sets equipped with the upper Vietoris topology. We obtain that a weak well-filtered space is core-compact iff it is locally compact. We also show that Ψχp-fine is equal to Ψωp-fine for a topological space under the condition of local compactness, which answers an open problem.