Dual discreteness of \(\Sigma \)-products and irreducibility of infinite products
type: Item
instance of: scholarly article
title: Dual discreteness of \(\Sigma \)-products and irreducibility of infinite products
author: Yasushi Hirata
author: Yukinobu Yajima
published in: Topology and its Applications
publication date: 2024-09-27T00:00:00Z
review text: All spaces considered in this paper are topological \(T_1\)-spaces, \(\mathbb{N}\) is an infinite countable discrete space, \(\omega\) is the first infinite cardinal, and \(\omega_1\) is the first uncountable cardinal.\N\NWe recall that a \textit{neighborhood assignment} for a space \((X,\tau)\) is a function \(U : X \rightarrow \tau\) such that \(x \in U(x)\) for each \(x \in X\). A space \(X\) is a \(D\)-\textit{space} if for every neighborhood assignment \(U\) for \(X\), there is a closed discrete subset \(D\) in \(X\) such that \(\{U(d) : d \in D\}\) covers \(X\). \(X\) is \textit{dually discrete} if for every neighborhood assignment \(U\) for \(X\), there is a discrete subset \(D\) of \(X\) such that \(\{U(d) : d \in D\}\) covers \(X\). A space \(X\) is an \(aD\)-\textit{space} if for every closed subset \(F\) of \(X\) and every open cover \(\mathcal{U}\) of \(X\), there is a closed discrete subset \(D\) of \(F\) such that for every \(d\in D\) one can assign \(U(d) \in \mathcal{U}\) with \(d\in U(d)\) such that \(\{U(d) : d \in D\}\) covers \(F\). A cover \(\mathcal{U}\) of a set \(S\) is \textit{minimal} if \(\mathcal{U}\setminus\{U\}\) does not cover \(S\) for any \(U \in\mathcal{U}\). A space \(X\) is \textit{irreducible} if every open cover of \(X\) has a minimal open refinement. We recall that every \(D\)-space is dually discrete and is an \(aD\)-space, and that the \(aD\)-spaces are irreducible.\N\NLet \(\left\{X_i:i\in I\right\}\) be a family of non-empty topological spaces and \(p=\{p_i:i\in I\}\) be a point in the product \(X_I\) of these spaces. The \(\Sigma\)-\textit{product} at \(p\), denoted by \(\Sigma(p)\), is the subspace of \(X_I\) that contains all points \(x=\{x_i:i\in I\}\) for which \(|\{i\in I:x_i \neq p_i\}|\le\omega\).\N\NThe authors of the paper under review proved in their previous paper [Topology Appl. 195, 297--311 (2015; Zbl 1329.54025)] that \(\mathbb{N}^{\omega_1}\) is not a \(D\)-space, and in this paper they show the following:\…
reviewed by: Ivan S. Gotchev
Mathematics Subject Classification ID: 54B10
Mathematics Subject Classification ID: 54D20
Mathematics Subject Classification ID: 54A25
zbMATH DE Number: 7919658
zbMATH Keywords: dually discrete
zbMATH Keywords: \(D\)-space
zbMATH Keywords: irreducible
zbMATH Keywords: \( \Sigma \)-product
zbMATH Keywords: product
MaRDI profile type: Publication
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DOI: 10.1016/J.TOPOL.2024.109032
description: scientific article; zbMATH DE number 7919658