Lattice-valued preordered sets as lattice-valued topological systems

This paper provides variable-basis lattice-valued analogues of the well-known results that the construct Prost of preordered sets, firstly, is concretely isomorphic to a full concretely coreflective subcategory of the category Top of topological spaces (which employs the concept of the dual of the specialization preorder), and, secondly, is (non-concretely) isomorphic to a full coreflective subcategory of the category TopSys of topological systems of S. Vickers (which employs the spatialization procedure for topological systems). Dualizing these results, one arrives at the similar properties of quasi-pseudo-metric spaces built over locales.