Proper actions on topological groups: Applications to quotient spaces

Details Proper actions on topological groups: Applications to quotient spaces Proper actions on topological groups: Applications to quotient spaces

2009-05-15

arxiv.org/abs/0905.2616v1

Mathematics

General Topology

11 pages

Scientific paper

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 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 ferred from X/G to X. Yet another application leads to the inequality dim X<= dim X/G + dim G for every paracompact group X and its locally compact subgroup G.

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