On almost sober spaces
type: Item
instance of: scholarly article
title: On almost sober spaces
zbMATH Open document ID: 1483.54013
author: Qidong Shan
author: Meng Bao
author: Xinpeng Wen
author: Xiaoquan Xu
published in: Topology and its Applications
publication date: 2021-11-26T00:00:00Z
review text: A topological \(T_0\) space \(X\) is called \textit{almost sober} iff each irreducible closed set has a join in the specialization order of \(X\). It is shown that almost sober spaces are closed under retracts, products and saturated subspaces. Other basic properties known from sober spaces are shown not to hold for almost sobriety, for example closed-hereditariness or the stability with respect to forming function spaces with the topology of pointwise convergence. Examples illustrate that almost sobriety of \(X\) is neither necessary nor sufficient for almost sobriety of its Smyth power space (the nonempty saturated compacta of \(X\) equipped with the upper Vietoris topology). The final section proves that the category of almost sober spaces with continuous mappings is not reflective in the category of \(T_0\) spaces and continuous mappings. The famous example of a non-sober Scott space [\textit{P. T. Johnstone}, Lect. Notes Math. 871, 282--283 (1981; Zbl 0469.06002)] functions prominently in this deduction.
reviewed by: Alexander Vauth
Mathematics Subject Classification ID: 54D99
Mathematics Subject Classification ID: 54B30
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 06B30
zbMATH DE Number: 7434156
zbMATH Keywords: almost sober space
zbMATH Keywords: hereditary property
zbMATH Keywords: retract
zbMATH Keywords: function space
zbMATH Keywords: Smyth power space
zbMATH Keywords: reflection
MaRDI profile type: Publication
full work available at URL: j.topol.2021.107896
OpenAlex ID: W3209340918
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DOI: 10.1016/J.TOPOL.2021.107896
description: scientific article; zbMATH DE number 7434156