$O_1$-convergence in partially ordered sets

Sun, Tao ¹ ; Li, Qingguo ² ; Fan, Nianbai ³

¹ College of Mathematics and Physics, Hunan University of Arts and Science, Changde, Hunan 415000, P. R. China;College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, P. R. China
² College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, P. R. China
³ College of Computer Science and Electronic Engineering, Hunan University, Changsha, Hunan 410082, P. R. China

Journal of nonlinear sciences and its applications, Tome 12 (2019) no. 10, p. 634-643.

Résumé

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., $\bf207$ (2016), 156--166], in this paper, we further propose the concept of B-consistent $S^*$-doubly continuous posets and prove that the $O_1$-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 $O_1$-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.

Détail

DOI : 10.22436/jnsa.012.10.02

Classification : 54A20, 06A06
Keywords: \(O_1\)-convergence, B-topology, \(S^*\)-doubly continuous poset, B-consistent \(S^*\)-doubly continuous poset

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