ИСТИНА

One of the ways to construct extensions of a topological space X is to complete it with respect to uniformities on it. All Dieudonn\'e complete extensions of X are obtained in such a way, and completions over which the action a: GxX --> X can be continuously extended are called G-extensions. 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 ossibility 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 (has compact G-extension). Boundedness, uniform equicontinuity and quasiboundedness of actions are characterized as action's uniform continuity on the (piecewise) semi-uniform product. The notion of a semi-uniform product is introduced by J.Isbell. From this point of view the origin of different examples of action's extensions are explained. An example of a G-space that has no Dieudonn\'e complete G-extensions is constructed by A.Sokolovskaya. The first example of a G-space which is not G-Tychonoff was constructed by M.Megrelishvili.