ИСТИНА

Hilbert's fifth problem started the investigation of topological group's structure. Results of L.Pontrjagin on compact groups, H.Yamabe, D.Montgomery and L.Zippin on locally compact groups, E.Skljarenko on coset spaces of locally compact groups are just a few to be mentioned. They are based on two approximation approaches: 1) every neighborhood of group's unit contains a (normal, compact) subgroup such that the corresponding coset space (or quotient group) is a metrizable space (or Lie group); 2) decomposition of a group (coset space) into an analog of Pontrjagin's Lie series for compact groups. We shall discuss analogues of such approximations in case of actions of subgroups of products of Cech complete groups.