Date: Wed, 19 Oct 1994 18:08:33 +0100 (MET) From: Paul Johnson <johnson@king.math.muni.cz>
Dear categories,
In ``Stone spaces'', Johnstone notes that the assumption that (A) every coherent locale is spatial
is equivalent to
(B) the prime ideal theorem for distributive lattices.
Is it generally agreed that the assumption that
(C) the spatial part of every coherent locale has coherent topology
is also equivalent to (A) and (B) above?
In a hopeless reply to my own querie, it seems (after one sleepless night) less and less likely that (C) ==> (B). Ironically, since the trivial locale is coherent (and spatial), for a coherent locale A to have a non-coherent spatial part, it must have a point. Conversely, axiom (C) still appears quite strong, and I mean for tangible reasons: For example, (C) holds iff the passage from coherent spaces to distributive lattices has a left adjoint in coherent maps, thus identifying what I would call the ``constructively-spatial'' distributive lattices as reflective in all. But might (C) hold without assuming any choice??? Apparently, most of this battle can be fought at the level of Boolean algebras, but the reductions I have yet to have sorted out. Cheers, PBJ.