TITLE: Find two closed subsets or real numbers such that $d(A,B)=0$ but $A\cap B=\varnothing$
QUESTION [12 upvotes]: Here is my problem: 

Find two closed subsets or real numbers such that $d(A,B)=0$ but $A\cap B=\varnothing$.

I tried to use the definition of being close for subsets like intervals but I couldn't find any closed sets. Any hint? Thank you.

REPLY [5 votes]: In $\mathbb{R}^2$, a visual example is $A= \mathbb{R} \times \{0\}$ and $B= \{(x,e^x) \mid x \in \mathbb{R} \}$.