Spectral decompositions of spaces induced by spectral decompositions of acting groups (Q388009)

The paper is devoted to studying some classes of spaces with special actions of groups using the technique of compatible systems of maps and their corresponding inverse systems of spaces. If the acting groups have nice properties it is expected that some topological properties of the groups are transferred to spaces on which the groups act. Under certain conditions\(,\) a compatible system of maps of the acting group induces a compatible system of maps of the related space which is then a dense subset of the limit space of the associated inverse system of spaces. By this approach several results for pseudocompact \(G\)-spaces \(X\) are obtained. It is proved that a pseudocompact space \(X\) with a continuous \(d\)-open \((\)and totally bounded\()\) action of a Čech complete group \((\)a subgroup of a product of Čech complete groups\()\) is \(\kappa \)-metrizable. Furthermore\(,\) the Stone-Čech and the maximal equivariant \(G\)-compactification of \(X\) is an openly generated or Dugundji compactum. Particularly\(,\) a pseudocompact \((\)compact\() \) coset space of a Čech complete group \((\)a subgroup of a product of Čech complete groups\()\) is \(\kappa \)-metrizable \((\)an openly generated compactum\()\). In addition\(,\) it is also shown how a \(d\)-open continuous action of an \(\aleph _{0}\)-balanced group on a pseudocompact space \(X\) induces a \(d\)-open continuous action of an \(\aleph _{0}\)-bounded group on \(X\) .