Does there exist a metric space $(X,d)$ such that for subsets $A$ and $B$, define $ d(A,B)= inf \{d(a,b): a \in A, b \in B \}$

Question

Does there exist a metric space $(X,d)$ such that for subsets $A$ and $B$, define $ d(A,B)= inf \{d(a,b): a \in A, b \in B \}$, $\bar{A} \cap \bar{B} = \phi$, but $d(A,B) \leq 0 $

I think so but don't have an example.

Answer 1

You can construct an example with smooth closed sets in $\mathbb{R}^2$. It is enough to choose $A := \{(x,y)\in\mathbb{R}^2:\ y\leq 0\}$ and $B := \{(x,y)\in\mathbb{R}^2:\ y \geq e^x\}$.