The starting point of this paper is the interpretation of Grothendieck's topologies in the language of relational systems (cfr. ); this permits an arithmetical study of these topologies. By applying new techniques introduced here, we have achieved interesting results. These include a new characterization of Noetherian topological spaces, and further topological structures of local type (tangent sites) for which we have shown numerous properties. Finally, we have used the affine-envelope functor to illustrate some applications to the theory of schemes.