Continuous mappings on subspaces of products with the \( \kappa \)-box topology
type: Item
instance of: scholarly article
title: Continuous mappings on subspaces of products with the \( \kappa \)-box topology
zbMATH Open document ID: 1181.54017
author: Ivan S. Gotchev
author: W. W. Comfort
published in: Topology and its Applications
publication date: 2009-11-05T00:00:00Z
review text: The authors prove that under suitable assumptions, a continuous function \(f\) on a product space \(X\) endowed with a certain box topology depends only on a ``few'' coordinates, and study the existence of continuous extensions of \(f\) to spaces \(Y\) containing \(X\). The obtained results generalize some results of \textit{W.\ W.\ Comfort} and \textit{S.\ Negrepontis} [Chain conditions in topology, Cambridge Tracts in Mathematics 79 (2008; Zbl 1156.54002)] and \textit{M.\ Hušek} [Symp.\ math.\ 17, Anelli Funz.\ contin., Gruppi infin., Convegni 1973, 25--41 (1976; Zbl 0372.54005)]. If \(\kappa\) and \(\alpha\) are cardinals, we write \(\kappa \ll \alpha\) if \(\kappa < \alpha\), and for every \(\beta < \alpha\) and \(\lambda< \kappa\) we have \(\beta ^\lambda < \alpha\). Let \(X\) be a topological space. A family \(\{ A_i : i \in I\}\) of non-empty subsets of \(X\) is \textit{locally \(< \kappa\)} if there is an open cover \({\mathcal U}\) of \(X\) such that \(|\{i \in I : U \cap A_i \neq \emptyset\}| < \kappa\) for each \(U \in {\mathcal U}\). The space \(X\) is \textit{pseudo-\((\alpha, \kappa )\)-compact} if every locally \(< \kappa\) family \(\{U _i : i \in I \}\) of non-empty open sets satisfies \(| I | < \alpha\). For every \(\{ X_i : i \in I\}\) and \(J \subseteq I\) we write \(X_J = \prod _{i \in J} X_i\). When each \(X_i\) is a topological space, \((X_I)_{\kappa }\) denotes \(X_I\) endowed with the \(\kappa\)-box topology, i.e.\ for which \(\{\prod _{i \in I} U_i : U_i \subseteq X_{i} ~(i \in I)\text{ are open}, |\{i \in I : U_{i} \neq X_{i}\}| < \kappa \}\) is a base. If \(Y\) and \(Z\) are topological spaces, we denote by \(C (Y , Z )\) the set of continuous functions from \(Y\) into \(Z\), and \(\Delta_{Z} = \{(z,z) : z \in Z\} \subseteq Z \times Z\). We cite three representative results from the paper. Theorem. Let \(\omega \leq \kappa \leq \alpha\) be cardinals such that either \(\kappa < \alpha\) or \(\alpha\) is regular. Let \(\{ X_i : i \in I\}\) b…
reviewed by: Tamás Mátrai
Mathematics Subject Classification ID: 54B10
Mathematics Subject Classification ID: 54D35
zbMATH DE Number: 5625458
zbMATH Keywords: continuous extension of function
zbMATH Keywords: product space
zbMATH Keywords: \(\kappa\)-box topology
zbMATH Keywords: \(\kappa\)-invariant set
zbMATH Keywords: \(\Sigma_{\kappa}\)-product
zbMATH Keywords: pseudo-\((\alpha, \kappa )\)-compact
zbMATH Keywords: Souslin number
zbMATH Keywords: calibre
MaRDI profile type: Publication
full work available at URL: j.topol.2009.04.015
OpenAlex ID: W2035159448
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DOI: 10.1016/J.TOPOL.2009.04.015
description: scientific article; zbMATH DE number 5625458