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Let $(X,d)$ be a metric space and $C\subset X$ be a compact subset. Let furthermore $G$ be a group that acts on $X$ proper and by isometries. Does there exist an $\epsilon >0 $ such that: Let $U=$ {$x\in X| d(x,C)<\epsilon$} and there are only finitely many $g\in G$ such that $g(U) \cap U \not= \emptyset$ ?

Cheers Helge

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does one need the assumption that $(X,d)$, is a metric space, where $d$ is a length metric ? –  Helge Jun 13 '13 at 18:12
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Being a length metric is irrelevant. On the other hand, you ought to assume the metric space is proper (closed balls are compact), particularly since you are assuming that the action is proper (the induced map $G \times X \to X \times X$ is a proper function). In that case what you ask for is true, and is a trivial consequence of the definitions. –  Lee Mosher Jun 13 '13 at 21:14
    
Helge: What Lee says is correct, such actions are usually called "metrically properly discontinuous". In your setting, you probably meant "properly discontinuous" (rather than just proper). Then the assertion is correct and is a nice exercise in point-set topology although not as easy as the one for "metrically proper" actions. –  Misha Jun 14 '13 at 3:07
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