??? coherent , T_{0} => sober ???
Dear Group, I would be very greatful if someone could help me to make clear the following situation (doubt) : 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" (1982) it is written (II.3.4) that a space X is coherent if it is sober and \Omega(X) is a coherent local and in the corollary (Stone's representation theorem for distributive lattices) he writes The category DLat is dual to the category CohSp of coherent spaces and coherent maps between them.
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
I have looked around and I didn't find that result. Am I misunderstanding something ? I'm finishing my MSc. thesis on "Behavior and State" and for the work I'm developing a result like the above would be very helpful. So, I would be very greatful if you could help me. Thanks in advance for time and trouble, Maria Joao Frade ------------------------------------------------------ Maria Joao Gomes Frade e-mail: mjf@di.uminho.pt Assistant Lecturer and MSc. student Departamento de Informatica, Universidade do Minho
I'm afraid this was a slip of the pen on my part. I should have included sobriety in the definition of coherent space (and then omitted T_0 in 2.3.3(iii)). This is what comes of trying to be ``helpful'' ... I included this material as general background only. Theorem 2.3.3 is never used in the remainder of the paper. Samson Abramsky
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.
participants (3)
-
Maria Joao Frade -
Samson Abramsky -
Steven Vickers