ИСТИНА

The investigation of action's extensions started with the following compactification problem: whether any G-space is G-Tychonoff (has compact G-extension)? This problem is satisfactory solved. J.de Vries characterized G-Tychonoff spaces using uniform structures and a concept of bounded action. Sufficient condition (quasiboundedness of action) when the action can be extended over the completion of a phase space was introduced by M.Megrelishvili.Quasibounded actions generalize both bounded and uniformly equicontinuous ones. Moreover, quasiboundedness also guarantees the possibility of action's extension over the completion of the acting group in two-sided uniformity. The existence of uniformities on a G-space with respect to which the action is quasibounded characterizers the case when the G-space is G-Tychonoff. Boundedness, uniform equicontinuity and quasiboundedness of actions are characterized as action's uniform continuity on the (piecewise) semi-uniform product. From this point of view the origin of different examples of action's extensions are explained. In the study of semi-lattices of compact G-extensions results and examples when a semi-lattice has a minimal or the smallest element are presented. Compact G-extensions of h-homogeneous spaces, in particular of rationals, are discussed.