Simultaneously reflective and coreflective full subconstructs of stratified L-topological spaces are concretely reflective and coreflective

Let L be a completely distributive lattice. A stratified L-topology on a set X is a subfamily of L-subsets of X which is closed with respect to arbitrary suprema and finite infinima, and contains all the constants. In this paper, it is shown that every simultaneously reflective and coreflective full subconstruct of stratified L-topological spaces is necessarily concretely reflective and coreflective. In other words, every such subconstruct is necessarily both initially and finally closed. As an application, it is demonstrated that the construct of bitopological spaces has exactly 4 simultaneously reflective and coreflective full subconstructs.