A COMPACT SPACE IS NOT ALWAYS SI-COMPACT

In this paper, 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 -lattice, that is, a complete lattice L is called a T -lattice if for any x ∈ L \ {1L}, ↑x \ {x} ∈ Q(L). We prove that (1) if P and Q are quasicontinuous domains or T -lattices, P is order isomorphic to Q iff (Q(P ),⊇) is order isomorphic to (Q(Q),⊇); and (2) for a compact space X, if the Scott space of the open set lattice O(X) is sober, then X is SI-compact. Finally, using the result (2) and the Isbell example of a non-sober complete lattice, we present a compact sober space which is not SI-compact, thus give a positive answer to an open question posed by Zhao and Ho.