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On topological spaces that have a bounded complete dcpo model. (English) Zbl 1427.06001

Summary: 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 previously proved by Xi and Zhao that every \(T_1\) space has a dcpo model. It is, however, still unknown whether every \(T_1\) 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 first show that the set of natural numbers equipped with the co-finite topology does not have a bounded complete dcpo model and 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 of the nonempty closed compact subsets of the given space.

MSC:

06B35 Continuous lattices and posets, applications
06B30 Topological lattices
54A05 Topological spaces and generalizations (closure spaces, etc.)
54D10 Lower separation axioms (\(T_0\)–\(T_3\), etc.)
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