Lattice-equivalence of convex spaces

Abstract

The links between order and topology have been extensively studied. The objective of this paper is to study the similar links between convex spaces and order structures via the lattices of convex sets. In particular, we prove a characterization of convex spaces uniquely determined by means of their lattices of convex sets, they are precisely sober and $$S_D$$ . One adjunction between the category of convex spaces and that of continuous lattices is constructed, revealing the connection between convex structures and domain theory.