2016/07/13 by Dongsheng Zhao, Zhao, Dongsheng, Luoshan Xu +1
Computer Science · Decision Sciences · Mathematics · #06B30 #06B35 #54C35 #Advanced Algebra and Logic #FOS: Mathematics #Fuzzy and Soft Set Theory #General Topology (math.GN) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.1607.03576
openalex publication_date 2016/07/13 · openalex created_date 2022/09/12 · openalex updated_date 2026/07/28
By Thron, a topological space X has the property that C(X) isomorphic to C(Y) implies X is homeomorphic to Y iff X is sober and TD, where C(X) and C(Y) denote the lattices of closed sets of X and T0 space Y, respectively. When we consider dcpos (directed complete posets) equipped their Scott topologies, a similar question arises: which dcpos P have the property that for any dcpo Q, Cσ(P) isomorphic to Cσ(Q) implies P is isomorphic to Q (such a dcpo P will be called Scott closed set lattice faithful, or SCL-faithful in short)? Here Cσ(P) and Cσ(Q) denote the lattices of Scott closed sets of P and Q, respectively. Following a characterization of continuous (quasicontinuous) dcpos in terms of Cσ(P), one easily deduces that every continuous (quasicontinuous) dcpo is SCL-faithful. Note that the Scott space of every continuous (quasicontinuous) dcpo is sober. Compared with Thron's result, one naturally asks whether every SCL-faithful dcpo is sober (with the Scott topology). In this paper we shall prove that some classes of dcpos are SCL-faithful, these classes contain some dcpos whose Scott topologies are not bounded sober. These results will help to obtain a complete characterization of SCL-faithful dcpos in the future.