Discretization in 2D and 3D Orders

Among the different discretization schemes that have been proposed and studied in the literature, the supercover is a very natural one, and furthermore presents some interesting properties. On the other hand, an important structural property does not hold for the supercover in the classical framework: the supercover of a straight line (resp. a plane) is not a discrete curve (resp. surface) in general. We follow another approach based on a different, heterogenous discrete space which is an order, or a discrete topological space in the sense of Paul S. Alexandroff. Generalizing the supercover discretization scheme to such a space, we prove that the discretization of a plane in R3 is a discrete surface, and we prove that the discretization of the boundary of any closed convex set X is equal to the boundary of the discretization of X.