Weak well-filtered spaces and coherence

It was shown in [6] that a well-filtered dcpo L is coherent in its Scott topology if and only if for every x,y∈L, ↑x∩↑y is compact in the Scott topology. We generalize this result to weak well-filtered posets which are defined in this paper. We obtain that every weak well-filtered poset is always a consistent dcpo. We separate the class of weak well-filtered dcpos from the well-filtered ones by showing that Johnstone's example of a non-sober dcpo belongs to the former but not the latter class.