Please use this identifier to cite or link to this item: http://hdl.handle.net/10497/23496
Title: 
Authors: 
Subjects: 
Scott topology
Sober space
Well-filtered space
Upper space
Issue Date: 
2021
Citation: 
Xu, X., Xi, X., & Zhao, D. (2021). A complete Heyting algebra whose Scott space is non-sober. Fundamenta Mathematicae, 252(3), 315-323. https://doi.org/10.4064/fm704-4-2020
Journal: 
Fundamenta Mathematicae
Abstract: 
We prove that (1) for any complete lattice 𝓛, the set 𝒟 (𝓛) of all non-empty saturated compact subsets of the Scott space of 𝓛 is a complete Heyting algebra (with the reverse inclusion order); and (2) if the Scott space of a complete lattice 𝓛 is non-sober, then the Scott space of 𝒟 (𝓛) is non-sober. Using these results and Isbell's example of a non-sober complete lattice, we deduce that there is a complete Heyting algebra whose Scott space is non-sober, thus giving an affirmative answer to a problem posed by Achim Jung. We also prove that 𝚊 𝒯₀ space is well-filtered iff its upper space (the set 𝒟 (𝓧) of all non-empty saturated compact subsets of X equipped with the upper Vietoris topology) is well-filtered, which answers another open problem.
URI: 
ISSN: 
0016-2736 (print)
1730-6329 (online)
DOI: 
Project number: 
RI 3/16 ZDS
Grant ID: 
NSFC(11661057)
20161BAB2061004
20192ACBL20045
NSFC (11361028)
NSFC (61300153)
NSFC (11671008)
NSFC (11701500)
NSFC (11626207)
BK20170483
Funding Agency: 
National Natural Science Foundation of China
National Institute of Education, Singapore
File Permission: 
Open
File Availability: 
With file
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