Constructing Tychonoff \(G\)-spaces which are not \(G\)-Tychonoff
In 1978 it was shown by \textit{Jan de Vries} [Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 26, 275-280 (1978; Zbl 0378.54028)] that if \(G\) is a locally compact Hausdorff group then every completely regular Hausdorff \(G\)-space has a \(G\)-compactification, i.e., it can equivariantly be embedded into a compact Hausdorff \(G\)-space. The question on the existence of such compactifications remained open for a number of years for more general acting groups \(G\), until \textit{M. G. Megrelishvili} [Russ. Math. Surv. 43, No. 2, 177-178 (1988); translation from Usp. Mat. Nauk 43, No. 2(260), 145-146 (1988; Zbl 0664.54024)] had shown the existence of a \(G\)-space with \(G\) non-locally compact, that admits no \(G\)-compactifications. A rather natural question had immediately sprung to life: do there exist non-locally compact topological groups \(G\) sharing the property that every \(G\)-space can be equivariantly compactified? Such groups are called \(V\)-groups in the paper under review. The main result obtained here by the authors is the following: Everey \(\aleph_0\)-bounded \(V\)-group is locally precompact. We recall that according to \textit{I. I. Guran} [Sov. Math., Dokl. 23, 173-175 (1981); translation from Dokl. Akad. Nauk SSSR 256, 1305-1307 (1981; Zbl 0478.22002)] a topological group \(G\) is called \(\aleph_0\)-bounded if for every neighborhood \(U\) of unity of \(G\) there is a countable subset \(S\subset G\) such that \(SV=G\). Locally precompact groups are just subgroups of locally compact ones. The authors establish also the existence of cyclic non-\(V\)-groups. A characterization of locally compact groups in terms of `` \(G\)-normality'' \ is obtained.
scientific article; zbMATH DE number 1307977
scientific article