Countable frames: When they are continuous and Stone duals of Scott spaces
Xiaodong Jia, Xiaoyong Xi
Source abstract
Matthew de Brecht raised the question of whether countable frames are continuous lattices. We prove that the continuity of a countable frame implies the quasicontinuity of its corresponding spectrum in the dual specialization order, and further show that this question admits a positive answer if the frame’s spectrum is a T 1 T_1 space or a Scott space. In general, we confirm the existence of non-continuous countable frames. This work also partially addresses an open problem posed by Jimmie Lawson and Michael Mislove in 1990, which concerns the characterization of when the spectra of spatial frames are Scott spaces.
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