偏序集上的测度拓扑和全测度

Measurement Topology and Full Measure on Posets

在偏序集上引入测度拓扑和全测度概念,研究其性质以及与其它内蕴拓扑间的众多关系。主要结果有:连续偏序集的测度拓扑实际上是由其上的任一全测度所决定且可由它的定向完备化上的测度拓扑和全测度分别限制得到;当连续偏序集还是D om a in时,其上的测度拓扑与μ拓扑一致;连续偏序集有可数基当且仅当其上的测度拓扑是可分的;一个网如果测度收敛则存在最终上确界;任一ω连续偏序集上都存在全测度。

The new concepts of the measurement topology and full measure on posets are introduced. Some basic properties of them and relations with other intrinsic topologies are given. The main results are： （1） The measurement topology on a continuous poset is actually determined by any full measure for the poser, and can also be obtained by restricting the measurement topology for the directed completion of the poser to this poser. （2） If the continuous poser is also a continuous dcpo, then the measurement topology of the poser is exactly theμ topology in the sense of K. Martin. （3） A continuous poset has a countable basis iff the measurement topology of it is separable. （4） If a net in a continuous poset convergences to a point in the measurement topology, then the net has eventually a supremum of that point. （5） Any ω continuous poser has a full measure on it.