Well-filtered spaces and their dcpo models

Abstract

A topological space X is called well-filtered if for any filtered family $\mathcal{F}$ of compact saturated sets and an open set U, ∩ $\mathcal{F}$ ⊆ U implies F ⊆ U for some F ∈ $\mathcal{F}$. Every sober space is well-filtered and the converse is not true. A dcpo (directed complete poset) is called well-filtered if its Scott space is well-filtered. In 1991, Heckmann asked whether every UK-admitting (the same as well-filtered) dcpo is sober. In 2001, Kou constructed a counterexample to give a negative answer. In this paper, for each T1 space X we consider a dcpo D(X) whose maximal point space is homeomorphic to X and prove that X is well-filtered if and only if D(X) is well-filtered. The main result proved here enables us to construct new well-filtered dcpos that are not sober (only one such example is known by now). A space will be called K-closed if the intersection of every filtered family of compact saturated sets is compact. Every well-filtered space is K-closed. Some similar results on K-closed spaces are also proved.