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

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Antonyan, Sergey A.

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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 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.

Comment: In the proof of Proposition 3.1 of the previous version there is a small gap. To correct the gap, at the end of the proof (now Proposition 3.2) one should just reference to a newly added Lemma 3.1 for the fact that Ux is a G-small set. Results unchanged. arXiv admin note: substantial text overlap with arXiv:1103.1407

Comment: In the proof of Proposition 3.1 of the previous version there is a small gap. To correct the gap, at the end of the proof (now Proposition 3.2) one should just reference to a newly added Lemma 3.1 for the fact that Ux is a G-small set. Results unchanged. arXiv admin note: substantial text overlap with arXiv:1103.1407

##### Keywords

Mathematics - General Topology, Mathematics - Geometric Topology, 22A05, 54H11