Compactification of G-spaces revisited

We discuss compactifications of $G$--spaces from a new point of view that completely differs from our earlier approaches. From a topologist's point of view, this new approach is more natural than the previous ones. In addition, it enables a unified discussion of the compactification and the linearization problem for $G$--spaces (which we shall discuss in a subsequent report). The central idea is to get a sufficiently large 'natural' family of elementary compact $G$--spaces which can play the same role as the interval $clint$ in ordinary topology