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 ⊂ X ; in particular, the restriction of the orbit projection X → X / G to F is a perfect surjective map F → 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 / G . We also show that every topological group X, endowed with the natural action of any locally compact subgroup G ⊂ X , is a uniformly proper G-space.