## Rocky Mountain Journal of Mathematics

### 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 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

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

#### 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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