Notes on the products of the lower topology and Lawson topology on posets

摘要

In this paper, we investigate the relation between the lower topology respectively the Lawson topology on a product of posets and their corresponding topological product. We show that (1) if S and T are nonsingleton posets, then Ω(S ×T) = Ω(S) × Ω(T) iff both S and T are finitely generated upper sets; (2) if S and T are nontrivial posets with σ(S) or σ(T) being continuous, then Λ(S × T) = Λ(S) × Λ(T) iff S and T satisfy property K, where for a poset L, Ω(L) means the lower topological space, Λ (L) means the Lawson topological space, and L is said to satisfy property K if for any x ∈ L, there exist a Scott open U and a finite F is contained in L with x ∈U is contained in ↑F.