QFS-domains and quasicontinuous domains

In this paper, we show that (1) for each QFS-domain L, L is an .QFS-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 x 1, x 2 . L and finite G 1, G 2 . L with G 1 . x 1, G 2 . x 2, there is a finite subset F . L such that . x 1. . x 2 .. F .. G 1. . G 2; (4) L is a QFS-domain iff L is a quasicontinuous domain and given any finitely many pairs {(F i , x i ): F i is finite, x i . L with F i . x i , 1 . i . n}, there is a quasi-finitely separating function . on L such that F i . .(x i ) . x i .