Faithfulness of Directed Complete Posets based on Scott Closed Set Lattices

Abstract: 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 $T_D$, where $C(X)$ and $C(Y)$ denote the lattices of closed sets of $X$ and $T_0$ 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 $Q$, $C_\sigma(P)$ isomorphic to $C_\sigma(Q)$ implies $P$ is isomorphic to $Q$ (such a dcpo $P$ will be called Scott closed set lattice faithful, or SCL-faithful in short)? Here $C_{\sigma}(P)$ and $C_{\sigma}(Q)$ denote the lattices of Scott closed sets of $P$ and $Q$, respectively. Following a characterization of continuous (quasicontinuous) dcpos in terms of $C_{\sigma}(P)$, 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.