Rocky Mountain Journal of Mathematics

On topological spaces that have a bounded complete DCPO model

Zhao Dongsheng and Xi Xiaoyong

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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 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.

Article information

Source
Rocky Mountain J. Math., Volume 48, Number 1 (2018), 141-156.

Dates
First available in Project Euclid: 28 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1524880885

Digital Object Identifier
doi:10.1216/RMJ-2018-48-1-141

Mathematical Reviews number (MathSciNet)
MR3795737

Zentralblatt MATH identifier
06866704

Subjects
Primary: 06B30: Topological lattices, order topologies [See also 06F30, 22A26, 54F05, 54H12] 06B35: Continuous lattices and posets, applications [See also 06B30, 06D10, 06F30, 18B35, 22A26, 68Q55] 54A05: Topological spaces and generalizations (closure spaces, etc.)

Keywords
Scott topology maximal point space bounded complete dcpo model CK-open set CK-filter defined space

Citation

Dongsheng, Zhao; Xiaoyong, Xi. On topological spaces that have a bounded complete DCPO model. Rocky Mountain J. Math. 48 (2018), no. 1, 141--156. doi:10.1216/RMJ-2018-48-1-141. https://projecteuclid.org/euclid.rmjm/1524880885


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