半连续dcpo

Semicontinuous dcpos

作为半连续格的推广,引入半素集和半连续dcpo的概念,并讨论半连续dcpo的性质,在半连续dcpo中得到类似于半连续格的一些主要结果.同时研究了dcpo的内蕴拓扑——半Scott拓扑、半Lawson拓扑,证明了上集U半Lawson开当且仅当U为半Scott开,下集U半Lawson闭当且仅当U为半Scott闭.最后研究了半连续映射,证明了若保序映射f半连续,则f关于半Scott拓扑是连续映射.

As a generalization of semicontinuous lattices, the concepts of serniprime sets and semi- continuous dcpos are introduced. Basic properties of semicontinuous dcpos are discussed. The main results of the theory of semicontinuous lattices are carried to semicontinuous dcpo. Moreover, some intrinsic topologies--the semi-Scott topology and the semi-Lawson topology on semicontinuous dcpos are investigated. It is proved that an upper set U is semi-Lawson open iff it is semi-Scott open; a lower set U is semi-Lawson closed iff it is semi-Scott closed. Finally the concept of semicon- tinuous mapping is introduced, it is showed that if f is order preserving and semicontinuous, then f is continuous with respect to the semi-Scott topologies.