On the dimension of spaces with a compact group of transformations

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.