Equivariant embeddings and \(\omega\)-bounded groups
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).
scientific article; zbMATH DE number 1028391