In this paper, we introduce the concepts of \({\mathcal {Q}^*}\) -determined spaces and \({\mathcal {Q}^*}\) -determined posets. We show that every \({\mathcal {Q}^*}\) -determined space is homeomorphic to the \({\mathcal {Q}^*}\) -spectrum of its Smyth hyperspace with the Scott topology. Similarly, every \({\mathcal {Q}^*}\) -determined poset is order isomorphic to the \({\mathcal {Q}^*}\) -spectrum of its Smyth powerdomain. So for any two \({\mathcal {Q}^*}\) -determined spaces or \({\mathcal {Q}^*}\) -determined posets X, Y with the Scott topology, X is homeomorphic to Y if and only if \({\mathcal {Q}}X\) is order isomorphic to \({\mathcal {Q}}Y\) . Moreover, based on the Hofmann-Mislove Theorem, we propose a counterexample for the following problem posed in Gierz et al. (2003): Is OFilt(S), the semilattice of all open filters of S, distributive for a distributive continuous semilattice S?
