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2018 On topological spaces that have a bounded complete DCPO model
Zhao Dongsheng, Xi Xiaoyong
Rocky Mountain J. Math. 48(1): 141-156 (2018). DOI: 10.1216/RMJ-2018-48-1-141


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.


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Zhao Dongsheng. Xi Xiaoyong. "On topological spaces that have a bounded complete DCPO model." Rocky Mountain J. Math. 48 (1) 141 - 156, 2018.


Published: 2018
First available in Project Euclid: 28 April 2018

zbMATH: 06866704
MathSciNet: MR3795737
Digital Object Identifier: 10.1216/RMJ-2018-48-1-141

Primary: 06B30 , 06B35 , 54A05

Keywords: bounded complete dcpo model , CK-filter defined space , CK-open set , maximal point space , scott topology

Rights: Copyright © 2018 Rocky Mountain Mathematics Consortium


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Vol.48 • No. 1 • 2018
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