Equivariant extensions

This paper introduces and studies equivariant versions of several fundamental topological extensions, namely, the Katetov extension (kappa X), the Hewitt realcompactification (upsilon X), the almost realcompactification (aX), and the Dieudonne completion (& micro;X) for topological transformation groups. The main results establish natural identifications between the H-orbit space of the equivariant extension of X and the corresponding extension of the orbit space X/H if G is a compact group acting on a Hausdorff (or Tychonoff) space X, and H is a closed normal subgroup of G. By setting H = G, we obtain (upsilon X-G) /G = upsilon (X/G), (& micro;X-G) /G = (& micro;X) /G = & micro; (X/G), (a(G)X) /G = a (X/G), and (kappa GX) /G = kappa (X/G). These results extend the classical theorem (beta X-G) /G = beta (X/G) for the Stone-Cech compactification.