Spaces determined by countably many locally compact subspaces
type: Item
instance of: scholarly article
title: Spaces determined by countably many locally compact subspaces
author: Jimmie D. Lawson
author: Xiaoquan Xu
published in: Topology and its Applications
publication date: 2024-08-01T00:00:00Z
review text: A topological space \(X\) is called an \(\ell c_\omega \)-space if there is an increasing sequence \((K_{n})_{n \in \mathbb{N}}\) of locally compact spaces such that their union equals \(X\), their inclusions \(K_{n} \rightarrow K_{n+1}\) are continuous, and a subset \(U\) of \(X\) is open if and only if \( U \cap K_{n} \) is open in \(K_{n}\) for each \( n \in \mathbb{N} \). In contrast with compactly-generated spaces, this property is preserved by forming finite products. To study posets equipped with the Scott topology, the authors introduce the notion of an \(\ell c_\omega \)-poset and other related concepts. It is proved that for \(\ell c_\omega \)-posets \(P\) and \(Q\), the product \( P \times Q \) is an \(\ell c_\omega \)-poset, the Scott space \(\Sigma P\) is an \(\ell c_\omega \)-space, and \( \Sigma (P \times Q) = \Sigma P \times \Sigma Q \). As an application, conditions are investigated under which the Scott space of a dcpo \(P\) is sober. A particular such one is that \(P\) is an \(\ell c_\omega \)-poset and \( \Sigma P \) is a strong \(d\)-space, which is a strengthened version of a \(d\)-space (monotone convergence space).
reviewed by: Alexander Vauth
Mathematics Subject Classification ID: 54D45
Mathematics Subject Classification ID: 54D50
Mathematics Subject Classification ID: 54D99
Mathematics Subject Classification ID: 06B35
Mathematics Subject Classification ID: 06F30
zbMATH DE Number: 7890716
zbMATH Keywords: locally compactly generated space
zbMATH Keywords: \( \ell c_\omega \)-space
zbMATH Keywords: Scott topology
zbMATH Keywords: product topology
zbMATH Keywords: sobriety
MaRDI profile type: Publication
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DOI: 10.1016/J.TOPOL.2024.108973
description: scientific article; zbMATH DE number 7890716