Proper actions on topological groups: Applications to quotient spaces

Sergey A. Antonyan
2010 Proceedings of the American Mathematical Society  
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 from
more » ... 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.
doi:10.1090/s0002-9939-2010-10504-x fatcat:7qcrqes2jvcqjixjny4ks2dnbm