A COMPACT SPACE IS NOT ALWAYS SI-COMPACT

We consider the poset Q(P) of all nonempty compact saturated subsets of the Scott space of a poset P, equipped with the reverse inclusion order. We also introduce the notion of T-lattices, that is, a complete lattice L is called a T-lattice if for any x is an element of L\ {1L}, up arrow x \ {x} is an element of Q(L). We prove that if P and Q are quasicontinuous domains or T-lattices, P is order isomorphic to Q if and only if (Q(P), superset of) is order isomorphic to (Q(Q), superset of). We also prove that for a compact space X, if the Scott space of the open set lattice O(X) is sober, then X is SI-compact. Using this result and the Isbell example of a non-sober complete lattice, we present a compact sober space which is not SI-compact. This gives a positive answer to an open problem posed by Zhao and Ho.