Continuous posets, prime spectra of completely distributive complete lattices, and Hausdorff compactifications

Abstract

In [24] we have established the equivalence of the following conditions (as far as necessary for our present purposes, terminology will be explained later in this Daper~ (I) X is a sober topological space which is "projective" with regard to the functor from the category So b o4 sober spaces and continuous maps into the category Poset of partially ordered sets (=posets) and isotone maps which makes a sober space Y into a partially ordered set by a~b iff a 6 el{b}. (Then the continuous maps become isotone maps.) (2) X is a continuous poset endowed with the Scott topok)q~. (3) X is a sober space which has an injective hull in the sense of B.Banaschewski [7] in the category T_o of T O-spaces and continuous maps. Here we want to add two further equivalent conditions (4) X is the (join-)prime spectrum of a completely dist-ributive complete lattice. (5) X is a sober space such that its lattice O(X) of open sets (ordered by inclusion) is completely distributive. Recently, J.D.Lawson [31] (5.6,5.9) has independently obtained the equivalence of (2),(4) and (5). From the analogous implication for continuous lattices ([43]II.I.13), (2) ==) (4) is easily derived. The proof of (4)-5 (2) given by Lawson requires a detailed analysis of the structure of the sub-poser of the prime elements of a completely dist-ributive complete lattice. We dispense with this by provina (4) ¢=¢ (3)~ A criterion of G.N.Raney [39] and, of course, B.Banaschewski's corollary 2 [7] on p.24o is required.