期刊论文

The connectedness of partially ordered sets equipped with related topologies

中文

湖南文理学院学报(自然科学版)

1672-6146

2022

34

4

1-4,30

10.3969/j.issn.1672–6146.2022.04.001

11801264:国家自然科学基金 2019JJ50505:湖南省自然科学基金

本校为第一机构

连通性是最重要的拓扑性质之一.为了研究序结构与拓扑结构之间的联系,本文研究了偏序集在赋予不同拓扑后的连通性.主要结果有:(1)Scott拓扑连通当且仅当偏序集P中不存在既为上集又为下集的非空真子集P1;(2)上(下)拓扑不连通当且仅当偏序集P中存在非空真子集P1,P2使得12P=P∪?P,且P1,P2均可表示为某个有限集的下(上)集;(3)考虑拓扑空间中子基的连通性,有上(下)拓扑空间连通当且仅当上(下)拓扑关于子基(Ψ)连通且空间中既开又闭的集合为既Ψ开又Ψ闭的集合;(4)任一有限偏序集P,上(下)拓扑空间是局部连通的.

Connectedness is one of the most important topological properties.In this paper,the connectedness of posets with different topologies is explored in order to study the relationship between ordered structure and topological structure.The main results are as follows:(1)Scott topology is connected if and only if there is no non-empty proper subset P1 that is both upper and lower in partially ordered sets P.(2)The upper(lower)topology is not connected if and only if there is a non-empty proper subset P1,P2 in the partially ordered sets P such that ...