On the action of the group of isometries on a locally compact metric space

Details On the action of the group of isometries on a locally compact metric space On the action of the group of isometries on a locally compact metric space

2009-02-02

arxiv.org/abs/0902.0319v3

Mathematics

General Topology

Shorter version submitted for publication

Scientific paper

In this short note we give an answer to the following question. Let $X$ be a locally compact metric space with group of isometries $G$. Let $\{g_i\}$ be a net in $G$ for which $g_ix$ converges to $y$, for some $x,y\in X$. What can we say 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 $g_{j}$ converges to $f$ pointwise on $C_x$ and $f(C_x)=C_{f(x)}$, where $C_x$ and $C_y$ denote the pseudo-components of $x$ and $y$ respectively. Applying this we give short proofs of the van Dantzig--van der Waerden theorem (1928) and Gao--Kechris theorem (2003).

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