PROPER ACTIONS ON TOPOLOGICAL GROUPS: APPLICATIONS TO QUOTIENT SPACES

Abstract

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 subset of X such that FG = 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 from X/G to X. Yet another application leads to the inequality dim X <= dim X/G+ dim G for every paracompact topological group X and a locally compact subgroup G of X having a compact group of connected components.