# $O_1$-convergence in partially ordered sets

Volume 12, Issue 10, pp 634--643 Publication Date: May 30, 2019       Article History
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### 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.

### Keywords

• $O_1$-convergence
• B-topology
• $S^*$-doubly continuous poset
• B-consistent $S^*$-doubly continuous poset

•  54A20
•  06A06

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