On topological spaces that have a bounded complete DCPO model

Abstract

A dcpo model of a topological space X is a dcpo (directed complete poset) P such that X is homeomorphic to the maximal point space of P with the subspace topology of the Scott space of P. It has been proved previously by X. Xi and D. Zhao that every T1 space has a dcpo model. It is, however, still unknown whether every T1 space has a bounded complete dcpo model (a poset is bounded complete if each of its upper bounded subsets has a supremum). In this paper we rst show that the set of natural numbers equipped with the co- nite topology does not have a bounded complete dcpo model, then prove that a large class of topological spaces (including all Hausdorff k-spaces) have a bounded complete dcpo model. We shall mainly focus on the model formed by all the nonempty closed compact subsets of the given space.