Metrizability of proper G-spaces and their orbit spaces

Abstract

For a locally compact group G, proper G-spaces in the sense of R. Palais are studied. One of our results states that each strongly metrizable proper G-space admits a G-invariant metric (compatible with its topology) provided G is an almost connected group. This extends several results about the existence of invariant metrics. We also prove that if the G-orbit space X/G of a proper G-space X is metrizable, then there exists a compact subgroup H of G such that the H-orbit space X/H is metrizable too. This is applied to show that IndX=dim⁡X in this case. Another result claims that if the orbit space X/G of a proper G-space X is a paracompact p-space, then X is also such.