ИСТИНА

We say that a topological group $G$ realizes homogeneity of a topological space X if there is a continuous transitive action of G on X. The investigation of spaces which homogeneity can be realized by groups from some fixed class of topological groups was initiated by A.~Arhangel'skii [1987]. In the talk we shall present results on simplification of acting groups preserving properties of actions: transitivity, being a coset space and preserving a fixed equiuniformity in case of a G-Tychonoff space. We shall give characterizations of topological spaces which are coset spaces of (separable) metrizable groups and complete metrizable (Polish) groups.