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