In article "Domain Theory in Logical Form" (1991) S. Abramsky says (definition 2.3.2) that
a space X is coherent if the \Omega(X) is a coherent local
and in theorem 2.3.3 (iii) he writes
CohSp \simeq CohLoc \simeq DLat^{op}
where CohSp is the category of coherent T_{0} spaces, and continuous maps which preserve compact-open subsets under inverse image.
But, in Johnstone's book "Stone Spaces" ... [standard defns omitted]
From the above definitions and results, I wonder if the following result is true: If the space X is T_{0} and \Omega(X) is a coherent local then the space X is sober
The standard definitions of coherent - or spectral - space (e.g. Hochster's in Trans AMS 142, 1969) clearly include sobriety as part of the definition (in Hochster: every closed irreducible subspace has a generic point), and that "T_0 + Omega X coherent" is insufficient. For a counterexample, let D be the Kahn domain of finite and infinite bit streams (i.e. order is prefix order, topology is Scott), and let D' be the subspace formed by removing a single infinite point. The inclusion D' -> D gives an isomorphism between Omega D' and Omega D, which is coherent, and D' is T_0. However, D' is not sober, because it lacks the point that was taken out. I think therefore that Samson's Defn 2.3.2 ought to be modified to include sobriety as part of the definition, and that Theorem 2.3.3 needs mild rewording (e.g. delete "T_0"). Apart from that, I'd be very surprised if there's anything else in the whole paper that has to be changed as a result. Steve Vickers.