# Discrete subgroups of isometry group of proper metric space

Let $X$ be a proper metric space and consider its isometry group $\mathrm{ISO}(X)$ endowed with the compact-open topology. Let $G$ be a subgroup of $\mathrm{ISO}(X)$.

Consider the following conditions on $G$:

(1) $G$ acts properly on $X$, i.e. any two points $x$ and $y$ in $X$ have neighborhoods $U_x$ and $U_y$ such that there are only a finite number of group elements $g \in G$ with $g(U_x)$ meeting $U_y$.

(2) $G$ is a discrete subgroup of $\mathrm{ISO}(X)$.

(3) The orbit $Gx$ is a discrete subset of $X$ for all $x \in X$.

My question: Is (1) equivalent with (2) or is (2) equivalent with (3), or neither? Does anything change if one assumes also that $X$ is CAT(0) and/or $G$ acts cocompactly.

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Could you give some context for the question? Did it arise in your research, or are you just curious? As Misha says below, it looks a lot like a homework question. I'm going to vote to close, although the question could be suitable for MO with more context. –  HJRW Mar 7 '12 at 19:55
It was not a homework question, just a question I asked myself. But I see now that the answer is not very difficult. I was just forgetting about Arzela-Ascoli theorem. So thanks for the hint Misha. –  unknown Mar 8 '12 at 8:22