By uniformly discrete I mean there exists a $C > 0$ such that for all $x \neq y$ we have $d(x, y) \geq C$. By proper I mean the preimage of every closed ball is compact.
Are there any examples of such spaces which are uncountable?
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Consider $(\mathbb R,d)$ which: $d(x,y)=1+|x-y|$ for $x\neq y$. – hamid kamali Jun 08 '15 at 09:15
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@hamidkamali That one is not proper. – Jun 08 '15 at 17:59
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No, every such space is at most countable. Fix $a\in X$ and write $X$ as $\bigcup_{n=1}^\infty \overline{B}(a,n)$. Since $X$ is proper, $\overline{B}(a,n)$ is compact. But it is also discrete, therefore finite. (An infinite compact set would have a limit point, contradicting discreteness).