8 papers · 1 filter
Sobriety of Regular Open Algebras in Second-Countable T3 Spaces
Xiaoyong Xi, Chong Shen, Dongsheng Zhao
For a topological space X, let RO(X) be the complete Boolean algebra of regular open subsets of X, ordered by inclusion. We prove that, for every second-countable T3 space X, the S…
A Quasicontinuous Domain Without an Interval Retract
Chong Shen, Xiaoyong Xi, Dongsheng Zhao
We give negative answers to two questions of X.Xu concerning the occurrence of the unit interval in nonquasialgebraic quasicontinuous domains. We construct, directly from the binar…
The set of real numbers with the cocountable topology has a bounded complete dcpo model
Dongsheng Zhao, Chong Shen, Xiaoyong Xi
We construct a bounded complete dcpo P whose maximal point space is homeomorphic to the set R of all reals equipped with the cocountable topology, thereby answering an open problem…
The complete Boolean algebra of regular open sets in real line is not sober
Xiaoyong Xi, Chong Shen, Dongsheng Zhao
Whether the Scott space of the complete Boolean algebra $\RO(\bbR)$ of regular open subsets in the real line is sober was explicitly recorded as an open problem by Xu and Zhao and…
The answers to two problems on dcpo models
Chong Shen, Xiaoyong Xi, Dongsheng Zhao
A poset model of a topological space is a poset such that is homeomorphic to the maximal point space of (the set Max() of all maximal points of equipped with…
The answers to two problems on maximal point spaces of domains
Xiaoyong Xi, Chong Shen, Dongsheng Zhao
A topological space is domain-representable (or, has a domain model) if it is homeomorphic to the maximal point space $\mbox{Max}(P)$ of a domain (with the relative Scott topol…
Products of two sober dcpo's need not be sober
Hualin Miao, Xiaoyong Xi, Xiaodong Jia +2
We construct two dcpo's whose Scott spaces are sober, but the Scott space of their order product is not sober. This answers an open problem on the sobriety of Scott spaces. Meantim…
The set of maximal points of an -domain need not be a -set
Gaolin Li, Chong Shen, Kaiyun Wang +2
A topological space has a domain model if it is homeomorphic to the maximal point space $\mbox{Max}(P)$ of a domain . Lawson proved that every Polish space has an -domain…