偏序集的内蕴拓扑连通性

Connectedness of intrinsic topologies of partial ordered sets

从序与拓扑的交叉考虑,进一步研究偏序集在多种内蕴拓扑下的连通性和局部连通性.主要结果有:(1)一个偏序集是序连通的当且仅当它赋予Alexandrov拓扑是连通的,也当且仅当它赋予Scott拓扑是连通的;(2)每一偏序集赋予Alexandrov拓扑是局部连通的,每一偏序集赋予Scott拓扑是局部连通的;(3)如果拓扑空间的特殊化偏序集序连通,则该拓扑空间是连通的;(4)构造反例说明了存在偏序集赋予下拓扑后是连通空间,但该偏序集本身不是序连通的.

In this paper,the connectedness and local connectedness of partial ordered sets under various intrinsic topologies are further studied by considering the interactions of order and topology.The main results are as follows:(1)A partially ordered set is order-connected if and only if it is connected endowed with the Alexandrov topology if and only if it is connected endowed with the Scott topology.(2)Each partial ordered set endowed with the Alexandrov topology is locally connected,and every partial ordered set endowed with the Scott topology is also locally connected.(3)If the specialization order of a topological space is connected,then the topological space itself is connected.(4)A counterexample is constructed to show that there is a partial ordered set with the lower topology is connected,but the partial ordered set endowed with the Scott topology is not order-connected.