Dimension of proper G-spaces

In this paper, for G a locally compact group (or a Lie group), we study the relationship between the covering dimensions of a proper G-space X and its orbit space X / G . We prove also that dim X = Ind X for every proper G-space X with a metrizable orbit space provided that G is either pro-Lie or σ-compact or has a metrizable quotient group of connected components.