If $E$ and $F$ are disjoint closed subsets of a metric space $(X,d)$, then is $dist~(E,F) >0$ always? If $E$ and $F$ are disjoint closed subsets of a metric space $(X,d)$, then is $dist~(E,F) >0?$
My attempt:
Suppose $dist~(E,F)=0.$ 
Then $\exists~e \in E,f \in F$ such that $~\forall ~r>0,~d(e,f)<r.$
$\implies f$ is an accumulation for the set $E$ and $e$ is an accumulation point for $F$.
But, since, closed sets retain their accumulation points $\implies f \in E \bigcap F$ . But given that $E \bigcap F = \{\emptyset\}$ leading to a contradiction. 
Hence, $dist~(E,F)>0.$
Could someone please have a look at my proof and point out flaws , if any.
Thank you very much for your help!
 A: Consider $\mathbb R^2$ with standard metric and let $E=\{\,(x,y)\mid xy=1\,\}$, $F=\{\,(x,y)\mid xy=0\,\}$.
Your argument about $e,f$ being accumulation points is false. Instead of
$$ \exists e\in E\exists f\in F\forall r>0\colon d(e,f)<r$$
we only have
$$\forall r>0\exists e\in E\exists f\in F\colon d(e,f)<r $$
A: No. In fact, the conclusion

$\exists~e \in E,f \in F$ such that $~\forall ~r>0,~d(e,f)<r$

implies that $d(e, f) = 0$, and since $d$ is a metric, this is forces $e = f$ (and so that $X$ is a singleton).
The definition of distance between sets $E, F$ in a metric space $(X, d)$ is
$$d(E, F) := \inf \{d(e, f) : e \in F, f \in F \},$$
so $d(E, F) = 0$ is equivalent to the existence of sequences $e_k$ in $E$ and $f_k$ in $F$ such that $\lim d(e_k, f_k) = 0$, that is, that
$$\forall r > 0 \, \exists e \in E, f \in F : d(e, f) < r.$$

For an example, consider the metric space $$(\Bbb R_+, d),$$ where
$$d(x, y) := \log \left\vert \frac{x}{y} \right\vert ,$$ take $E$ to be the set of positive, even numbers and $F$ the set of positive, odd numbers, which are closed and disjoint. For any positive integer $k$ we have $2 k \in E$ and $2 k - 1 \in F$, but
$$\lim d(2k, 2k - 1) = \log \left\vert \frac{2 k}{2 k - 1} \right\vert = 0,$$ so $d(E, F) = 0.$
A: You can have $d(E,F)=0$.
Consider the real plane, $E=\{(x,y) \in \mathbb R^2 | x \ge 0, y \ge 0 \text{ and } xy \ge 1\}$ and $F$ the $x$-axis.
$P_n=(n,\frac{1}{n}) \in E$, $Q_n=(n,0) \in F$ for $n \ge 1$ and $\lim d(P_n,Q_n)=0$
