测度拓扑和交连续偏序集的刻画

Measurement Topology and Characterizations of Meet Continuous Posets

研究偏序集上的测度拓扑以及与其它内蕴拓扑间的若干关系,利用测度拓扑刻画了偏序集的交连续性。主要结果有:一个网如果测度收敛则存在最终上确界;偏序集上的测度拓扑实际上是由其上的任一全测度决定;拟连续偏序集上的测度拓扑是零维的;一个偏序集是交连续偏序集当且仅当其上的测度拓扑的开集的上集为Scott开集当且仅当它的D-完备化是交连续dcpo.

Some properties of the measurement topology on posets and relations with other intrinsic topologies are given. In terms of the measurement topology, meet continuity of posers are characterized. The main results are： （1） If a net in a poser converges to a point in the measurement topology, then the net has eventually a supremum of that point; （2） the measurement topology on a poset is actually determined by any full measure for the poset; （3） the measurement topology on a quasicontinuous poset is zero-dimensional; （4） a poset is meet continuous if and only if the upper sets of all open sets in the measurement topology are Scott open if and only if its D-completion is a meet continuous dcpo.