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Let X be a Hausdorff topological group and G a locally compact subgroup of X. We show that the natural action of G on X is proper in the sense of R. Palais. This is applied to prove that there exists a closed set F ⊂ X such that F G = X and the restriction of the quotient projection X → X/G to F is a perfect map F → X/G. This is a key result to prove that many topological properties (among them, paracompactness and normality) are transferred from X to X/G, and some others are transferred fromdoi:10.1090/s0002-9939-2010-10504-x fatcat:7qcrqes2jvcqjixjny4ks2dnbm