Invariant Uniformities on proper G-Spaces

S. A. Antonyan, S. de Neymet.


Let G be a locally compact Hausdorff group. A G-space is a topological space X equipped with a fixed action of G on X, i.e., a continuous map a: G ´ X ® X such that a(e, x) = x and a(g', a(g, x)) = a(g' g, x) for all x Î X, g, g' Î G and e the unity of G. A G-space X is called proper in the sense of R. Palais, if X is Tychonoff and each point of X has a neighborhood V such that for every point of X there is a neighborhood U with the property that the set <U, V> = {G Î g| gU ÇV¹ Æ} has compact closure in G. If G is a Lie group and X is a separable metrizable proper G-space then there exists a G-invariant metric on X, compatible with its topology (R. Palais, 1960). The question whether the separability of X is essential in Palais' theorem, still remains open. In this work we prove the existence of a compatible G-invariant uniformity on each proper G-space.