G-space; invariant metric; slice
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.
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