covering positively disjoint sets with finite balls I have a question if $X$ is a separable bounded metric space and $A$,$B\subset X$ are sets such that
$d(A,B)>0.$ 
Does there exists sets $A^{\prime}$ and $B^{\prime}$ such that each of them is
a finite union of open balls $A^{\prime}\cap B^{\prime}=\emptyset$, $A\subset
A^{\prime},$ and $B\subset B^{\prime}?$

If $X= \mathbb{R}$, $A=]- \infty,-1]$, $B=[1, \infty[$, and $d(x,y)=\frac{|x-y|}{1+|x-y|}$,it is false.
The balls of ($\mathbb{R},d)$ are the same that the balls of $\mathbb{R}$ with the usual metric. So $A$ can not be covered with a finite union of balls, without covering $B$.
