Orbit spaces and unions of equivariant absolute neighborhood extensors

Let G be a locally compact Hausdorff group. We study orbit spaces and unions of equivariant absolute neighborhood extensors (G-ANEs) in the category of all proper G-spaces that are metrizable by a G-invariant metric. We prove that if a proper G-space X is a G-ANE such that all the G-orbits in X are metrizable, then the G-orbit space X/G is an ANE. Equivariant versions of Hannerʹs theorem and Kodamaʹs theorem about unions of absolute neighborhood extensors are established. We also introduce the notion of a G-polyhedron and prove that if G is any compact group, then every G-ANR is arbitrary closely dominated by a G-polyhedron. Each G-polyhedron is a G-ANE.