QFS-domains and quasicontinuous domains

Abstract

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 ⊆ L with G1 ≪ x1, G2 ≪ x2, there is a finite subset F ⊆ L such that ↑ x1∩ ↑ x2 ⊆↑ F ⊆↑ G1∩ ↑ G2; (4) L is a QFS-domain 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 δ on L such that Fi ≪ δ(xi) ≪ xi.