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On the group of isometries on a locally compact metric space. (English) Zbl 1012.54046
Let $$(X,d)$$ be a locally compact metric space and let $$I(X,d)$$ be its group of isometries endowed with the topology of pointwise convergence. Let $$\Sigma(X)$$ denote the space of the connected components endowed with the quotient topology. The authors prove the following statements concerning the group $$I(X,d)$$ and its action on the space $$X$$: (1) If $$\Sigma(X)$$ is not quasicompact, then $$I(X,d)$$ need not be locally compact, nor act properly on $$X$$. (2) If $$\Sigma(X)$$ is quasicompact then (a) $$I(X,d)$$ is locally compact, (b) the action of it on $$X$$ is not always proper, and (c) the action of it is proper if $$X$$ is connected.

##### MSC:
 54H20 Topological dynamics (MSC2010) 37B99 Topological dynamics 54E15 Uniform structures and generalizations 54H15 Transformation groups and semigroups (topological aspects)
##### Keywords:
Ellis semigroup; isometry; pointwise convergence; proper action
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