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On the action of the group of isometries on a locally compact metric space. (English) Zbl 1229.54047
Let $$X$$ be a locally compact metric space and let $$G$$ be a group of isometries of $$X$$ endowed with the topology of pointwise convergence. In this case this topology coincides with the compact-open topology and $$G$$ is a topological group with it. An equivalence relation $$x\sim y$$ on $$X$$ is introduced defining $$x\sim y$$ when $$x$$ and $$y$$ can be connected by a finite sequence of intersecting open balls with compact closure. The equivalence classes are called pseudo-components and the space $$X$$ is said to be pseudo-connected if it has only one pseudo-component.
A continuous action of a topological group $$H$$ on a topological space $$Y$$ is said to be proper if the map $$H\times Y\to Y\times Y$$ with $$(g,x)\to (x,gx)$$ for $$g\in H$$, $$x\in Y$$, is proper, i.e. it is continuous, closed and the inverse image of a singleton is compact. As the main result the author gives a short proof of the following two theorems by D. van Dantzig and B. L. van Waerden [Abhandlungen Hamburg 6, 367–376 (1928; JFM 54.0603.04)] and S. Gao and A. S. Kechris [Mem. Am. Math. Soc. 766 (2003; Zbl 1012.54038)]:
If $$X$$ is connected then $$G$$ acts properly on $$X$$ and $$G$$ is locally compact.
If $$X$$ has finitely many pseudo-components then the group of all isometries of $$X$$ is locally compact. If $$X$$ is pseudo-connected, then the group acts properly on $$X$$.

##### MSC:
 54H15 Transformation groups and semigroups (topological aspects)
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