O1-convergence in partially ordered sets

Based on the introduction of notions of S*-doubly continuous posets and B-topology in [T. Sun, Q. G. Li, L. K. Guo, Topology Appl., 207 (2016), 156–166], in this paper, we further propose the concept of B-consistent S -doubly continuous posets and prove that the O1-convergence in a poset is topological if and only if the poset is a B-consistent S -doubly continuous poset. This is the main result which can be seen as a sufficient and necessary condition for the O1-convergence in a poset being topological. Additionally, in order to present natural examples of posets which satisfy such condition, several special sub-classes of B-consistent S*-doubly continuous posets are investigated.