By separable I mean, given any pair of disjoint closed sets, $A_1$ and $A_2$, a metric space $(X,d)$, there exists disjoint open sets $O_1$ and $O_2$ such that $A_1\subset O_1$ and $A_2\subset O_2$.
I can prove that for any $x\in A_1$, $x$ is separable from $A_2$, because it's easy to prove that $d(x,A_2)=\delta>0$. (If not, you can find a sequence in $A_2$ that converges to $x$, which contradicts with $A_2$ being closed.)
However it is still not obvious that $A_1$ and $A_2$ are separable because each time you find different $\delta$.