Bounded sets in spaces and topological groups
type: Item
instance of: scholarly article
title: Bounded sets in spaces and topological groups
zbMATH Open document ID: 0979.54037
DOI: 10.1016/S0166-8641(98)00098-4
published in: Topology and its Applications
publication date: 2002-02-18T00:00:00Z
review text: A subset \(A\) of a Tikhonov space \(X\) is C-compact (or relatively pseudocompact) in \(X\) if for every continuous real-valued function \(f\) on \(X\) the image \(f(A)\) is compact (or, every infinite system of open sets in \(X\) meeting \(A\) has an accumulation point in \(A\), resp.). Both concepts coincide in topological groups and are not finitely productive in the category of Tikhonov spaces. It is proved in the present paper that if \(A\) and \(B\) are C-compact (or relatively pseudocompact) in \(X\) and \(Y\), and one of the spaces \(X,Y\) is a topological group, then \(A\times B\) is C-compact (or relatively pseudocompact, resp.) in \(X\times Y\). From other results: 1.3: If \(A_i\) are C-compact in topological groups \(X_i\) then \(\prod A_i\) is C-compact in \(\prod X_i\). 3.3+3.4: The Hewitt-Nachbin realcompactification of an R-factorizable group is an R-factorizable group.
Mathematics Subject Classification ID: 54H11
Mathematics Subject Classification ID: 54B10
Mathematics Subject Classification ID: 54D35
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 54D60
Mathematics Subject Classification ID: 54C30
zbMATH DE Number: 1395045
zbMATH Keywords: topological group
zbMATH Keywords: product of spaces
zbMATH Keywords: boundedness
author: Salvador Hernández
author: Manuel Sanchis
author: Mikhail Tkachenko
reviewed by: M. Hušek
MaRDI profile type: Publication
full work available at URL: s0166-8641(98)00098-4
OpenAlex ID: W2099268318
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description: scientific article; zbMATH DE number 1395045