# The group of isometries of a locally compact metric space with one end

Manoussos A (2010) *Topology and its Applications* 158(18): 2876-2879.

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In this short note we give an answer to the following question. Let $X$ be alocally compact metric space with group of isometries $G$. Let $\{g_i\}$ be anet in $G$ for which $g_ix$ converges to $y$, for some $x,y\in X$. What can wesay about the convergence of $\{g_i\}$? We show that there exist a subnet$\{g_j\}$ of $\{g_i\}$ and an isometry $f:C_x\to X$ such that $g_{j}$ convergesto $f$ pointwise on $C_x$ and $f(C_x)=C_{f(x)}$, where $C_x$ and $C_y$ denotethe pseudo-components of $x$ and $y$ respectively. Applying this we give shortproofs of the van Dantzig--van der Waerden theorem (1928) and Gao--Kechristheorem (2003).

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Manoussos A. The group of isometries of a locally compact metric space with one end.

*Topology and its Applications*. 2010;158(18):2876-2879.Manoussos, A. (2010). The group of isometries of a locally compact metric space with one end.

*Topology and its Applications*,*158*(18), 2876-2879. doi:10.1016/j.topol.2010.09.008Manoussos, A. (2010). The group of isometries of a locally compact metric space with one end.

*Topology and its Applications*158, 2876-2879.Manoussos, A., 2010. The group of isometries of a locally compact metric space with one end.

*Topology and its Applications*, 158(18), p 2876-2879. A. Manoussos, “The group of isometries of a locally compact metric space with one end”,

*Topology and its Applications*, vol. 158, 2010, pp. 2876-2879. Manoussos, A.: The group of isometries of a locally compact metric space with one end. Topology and its Applications. 158, 2876-2879 (2010).

Manoussos, Antonios. “The group of isometries of a locally compact metric space with one end”.

*Topology and its Applications*158.18 (2010): 2876-2879.
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arXiv 0902.0319