December 2022 A COMPACT SPACE IS NOT ALWAYS SI-COMPACT
Zhengmao He, Kaiyun Wang
Rocky Mountain J. Math. 52(6): 2041-2051 (December 2022). DOI: 10.1216/rmj.2022.52.2041

Abstract

We consider the poset 𝒬(P) of all nonempty compact saturated subsets of the Scott space of a poset P, equipped with the reverse inclusion order. We also introduce the notion of T-lattices, that is, a complete lattice L is called a T-lattice if for any xL{1L}, x{x}𝒬(L). We prove that if P and Q are quasicontinuous domains or T-lattices, P is order isomorphic to Q if and only if (𝒬(P),) is order isomorphic to (𝒬(Q),). We also prove that for a compact space X, if the Scott space of the open set lattice 𝒪(X) is sober, then X is SI-compact. Using this result and the Isbell example of a non-sober complete lattice, we present a compact sober space which is not SI-compact. This gives a positive answer to an open problem posed by Zhao and Ho.

Citation

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Zhengmao He. Kaiyun Wang. "A COMPACT SPACE IS NOT ALWAYS SI-COMPACT." Rocky Mountain J. Math. 52 (6) 2041 - 2051, December 2022. https://doi.org/10.1216/rmj.2022.52.2041

Information

Received: 8 November 2020; Revised: 27 October 2021; Accepted: 30 October 2021; Published: December 2022
First available in Project Euclid: 28 December 2022

MathSciNet: MR4527008
zbMATH: 07639788
Digital Object Identifier: 10.1216/rmj.2022.52.2041

Subjects:
Primary: 06B35
Secondary: 06B30 , 54A05

Keywords: compact saturated set , quasicontinuous domain , scott topology , SI-compactness , T-lattice

Rights: Copyright © 2022 Rocky Mountain Mathematics Consortium

Vol.52 • No. 6 • December 2022
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