Birkhoff's order-convergence in partially ordered sets

By introducing new approximation relations on posets, the notion of S⁎S⁎-doubly continuous posets is proposed and its relationship with doubly continuous posets is presented. Aslo the B-topology on posets is defined and the special properties of the B-topology on S⁎S⁎-doubly continuous posets are studied. Then, a sufficient and necessary condition for the Birkhoff's order-convergence in a poset being topological is obtained.