Metrics in locally compact groups

Raimond A. Struble

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Invariant metrics on $G$ -spaces

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Let $X$ be a $G$ -space such that the orbit space $X / G$ is metrizable. Suppose a family of slices is given at each point of $X$ . We study a construction which associates, under some conditions on the family of slices, with any metric on $X / G$ an invariant metric on $X$ . We show also that a family of slices with the required properties exists for any action of a countable group on a locally compact and locally connected metric space.