Weak Domain Models of T1 spaces

The main objective of this paper is to study some aspects of weak domains. We first show that every meet continuous weak domain is a domain. Then we prove that a dcpo P is exact iff the weakly way-below relation is the smallest approximating w-auxiliary relation. It is then shown that for each T1 space, Zhao and Xi's dcpo model is a weak algebraic domain, and hence a weak domain. As a consequence, we have that a weak algebraic domain need not be well-filtered and that every T1 space has a weak domain model, which strengthens a result of Mashburn.