Equivariant embeddings and \(\omega\)-bounded groups
type: Item
instance of: scholarly article
title: Equivariant embeddings and \(\omega\)-bounded groups
zbMATH Open document ID: 0882.54031
published in: Moscow University Mathematics Bulletin
publication date: 1997-10-08T00:00:00Z
review text: A topological group is called \(\omega\)-bounded if for every one of its unit neighbourhoods, a countable number of translates covers the group. The author deduces two characterizations of \(\omega\)-boundedness: thus, \(G\) is \(\omega\)-bounded if and only if every equicontinuous compact subset of \(C(G)\) is metrizable. A dynamical system \((G,X)\) is called \(G\)-Tikhonov if \(G\) and \(X\) are Hausdorff spaces, and separation of points and closed sets in \(X\) can be effected by functions satisfying a certain condition of uniform continuity with respect to the action of \(G\). In analogy to the classical result by which a Tikhonov space is homeomorphic to a subspace of some power of an interval, it is shown that for an \(\omega \)-bounded \(G\), every \(G\)-Tikhonov space is embeddable into a product of \(G\)-spaces which are compact convex metrizable \(G\)-invariant subsets of certain function spaces on which \(G\) operates by argument translation. The number of factors needed is equal to the topological weight of \(X\) (if the latter is infinite).
Mathematics Subject Classification ID: 54H11
Mathematics Subject Classification ID: 54C25
zbMATH DE Number: 1028391
zbMATH Keywords: dynamical system
zbMATH Keywords: uniform continuity
zbMATH Keywords: Tikhonov space
author: Sergey A. Antonyan
reviewed by: Peter Flor
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1028391