Topological predomains and qcb spaces are not closed under sobrification

In (Simpson 2003) A. Simpson introduced the category PreDom of topological predomains as a framework for denotational semantics containing also most classical spaces, namely all countably based T0 spaces. Countably based T0 spaces are isomorphic to subspaces of Pω where the latter is endowed with the Scott topology. A qcb space (as introduced in (Menni and Simpson 2002)) is a T0 space which appears as quotient of a countably based T0 space. In (Schröder 2003) qcb spaces have been characterized as those sequential spaces X for which there exists a countable pseudobase, i.e. a countable subset B of P(X) such that for every converging sequence (xn) → x and open neighbourhood U of x there exists a B ∈ B with B ⊆ U and (xn) eventually in B. We write QCB for the category of qcb spaces and continuous maps. It has been stated in (Simpson 2003) and proved in (Battenfeld 2004) that QCB is equivalent to ExPerΣ(Pω), the category of Σ-extensional per’s over Scott’s Pω (where Σ is the Sierpinski space). A topological predomain is a qcb space X which is also a monotone convergence space, i.e. X is a directed complete partial order (dcpo) w.r.t. the specialization order vX and every U ∈ O(X) is Scott open (w.r.t. vX).