Completely Scott closed set and its applications

In this paper, we propose a concept of completely Scott closed sets and use it to study links between convex spaces and continuous lattices. Firstly, we take three equivalent approaches to construct a convex space from a continuous lattice. Secondly, we construct an adjunction between the category of convex spaces and the opposite category of continuous lattices via completely Scott closed sets. This adjunction exactly induces the concept of sober convex spaces which gives rise to a categorical duality between them and algebraic lattices. Finally, we prove that completely Scott closed sets form a monad over the category of convex spaces and obtain an isomorphism between the category of sober convex spaces and the Eilenberg–Moore category of this monad.