Given a locale A, one may endow the set of points Loc(1,A) with the relative topology enherited from A, (and thereby produce the "spatial part of A"). Or, viewing Loc(1,A) as a poset with directed sups (in the "specialization" ordering), one may equip Loc(1,A) with the Scott topology. I have convinced myself that for arbitrary A, the Scott topology is at least as fine as the relative topology. Are there conditions on the locale A which characterize the coincidence of these two ways one might topologize its set of points? Paul
"The Scott topology is at least as fine as the relative topology." This is Corollary 7.3.2 in my book "Topology via Logic". "Are there conditions characterizing the coincidence of the two topologies on the points of a locale A?" I suspect that results such as Johnstone's "Scott is not always sober" might make exact characterizations difficult. Also, as stated in the original question, the characterization would have to include all locales without any points, so perhaps the class is not a terribly natural one to consider. Of course, a useful sufficient condition is for A to be completely distributive as a frame, this corresponding spatially to Scott topologies on continuous posets. (See Johnstone's "Stone Spaces".) Steve Vickers
participants (2)
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P. B. Johnson -
Steven John Vickers