Geodesic
On~the~dimension of~spaces with a~compact group of~transformations
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 31 (1976) no. 5, pp. 128-137

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The main result of this paper is as follows: {\it If a compact group $K$ acts continuously on a normal space $X$ so that the orbit space $X/K$ is metrizable, then $\dim X=\operatorname{Ind}X$}. Particular cases of spaces on which a compact group acts continuously with a metrizable orbit space are locally compact groups and their quotient spaces and also almost metrizable (in particular, Čech-complete) groups [5] and their quotient spaces. All the spaces we consider are assumed to be Hausdorff, and $X$ completely regular. All subgroups that occur are closed and all maps are continuous. 
@article{RM_1976_31_5_a11,
     author = {B. A. Pasynkov},
     title = {On~the~dimension of~spaces with a~compact group of~transformations},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {128--137},
     publisher = {mathdoc},
     volume = {31},
     number = {5},
     year = {1976},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/RM_1976_31_5_a11/}
}
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B. A. Pasynkov. On~the~dimension of~spaces with a~compact group of~transformations. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 31 (1976) no. 5, pp. 128-137. http://geodesic.mathdoc.fr/item/RM_1976_31_5_a11/