Uniformly proper actions

For a locally compact Hausdorff group G we introduce the notion of a uniformly proper G-space. We prove that a uniformly proper G-space X   admits a closed fundamental set F⊂XF⊂X; in particular, the restriction of the orbit projection X→X/GX→X/G to F   is a perfect surjective map F→X/GF→X/G. This is a key result to prove the existence of a compatible G-invariant metric on a uniformly proper metrizable G-space. Many topological properties, among them metrizability and paracompactness, are transferred from X   to X/GX/G. We also show that every topological group X  , endowed with the natural action of any locally compact subgroup G⊂XG⊂X, is a uniformly proper G-space.