Well-filtered spaces, compactness, and the lower topology

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It has long been known that a locally compact well-filtered T0space is sober. The question has been asked whether locally compact could be weakened to core compact, and we give a positive answer to this question in this paper. We next turn to a detailed study of the lower topology of a partially ordered set, where the partial order may be the order of specialization of a T0-space. After deriving some of its elementary properties related to compactness, we consider T0-spaces with the distinguished property that every set closed in the lower topology is compact in the lower topology. A key result is that a dcpo equipped with the Scott-topology satisfying this distinguished property is well-filtered, and the result generalizes to a T0space with its topology determined by its directed subsets. We also derive a converse of this result.

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Houston Journal of Mathematics

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