QFS-domains and Quasicontinuous Domains

In this paper, we show that （1） for each QFS-domain L, L is an ωQFS-domain iff L has a countable base for the Scott topology; （2） the Scott-continuous retracts of QFS-domains are QFS- domains; （3） for a quasicontinuous domain L, L is Lawson compact iff L is a finitely generated upper set and for any x1, x2 ∈ L and finite G1, G2 C L with G1 〈〈 x1, G2 〈〈 x2, there is a finite subset F C L such that ↑ x1 x2 G2; （4） L is a QFS-d0main iff L is a quasicontinuous domain and given any finitely many pairs {（Fi, xi） ： Fi is finite, xi ∈ L with Fi 〈〈 xi, 1 ≤i ≤n}, there is a quasi-finitely separating function 5 on L such that Fi 〈〈 δ（xi） 〈〈 xi.