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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 \(X\).

54H15 Transformation groups and semigroups (topological aspects)
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