The B-topology on S*-doubly quasicontinuous posets

The notions of os-convergence and S*-doubly quasicontinuous posets are introduced, which can be viewed as common generalizations of Birkhoff's order-convergence and S*-doubly continuous posets, respectively. We first consider the relationship between os-convergence and B-topology and show that the topology induced by o(s)-convergence according to the standard topological approach is the B-topology precisely. Then, the topological characterization for the S*-doubly quasicontinuity is presented. It is proved that a poset is S*-doubly quasicontinuous iff the poset equipped with the B-topology is locally hyperclosed iff the lattice of all B-open subsets of the poset is hypercontinuous. Finally, the order theoretical condition for the os-convergence being topological is given and the complete regularity of B-topology on S*-doubly quasicontinuous posets is explored.

Automated summary

The concepts of os-convergence and S*-doubly quasicontinuous posets were introduced as generalizations of Birkhoff's order-convergence and S*-doubly continuous posets. The relationship between os-convergence and B-topology was explored, along with the topological characterization for S*-doubly quasicontinuity. The order theoretical condition for os-convergence being topological and the complete regularity of B-topology on S*-doubly quasicontinuous posets were also discussed.