\(O_1\)-convergence in partially ordered sets

Volume 12, Issue 10, pp 634--643 http://dx.doi.org/10.22436/jnsa.012.10.02
Publication Date: May 30, 2019 Submission Date: April 03, 2017 Revision Date: February 27, 2019 Accteptance Date: April 30, 2019

Authors

Tao Sun - 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. Qingguo Li - College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, P. R. China. Nianbai Fan - College of Computer Science and Electronic Engineering, Hunan University, Changsha, Hunan 410082, P. R. China.


Abstract

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.


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