question about distance of sets

Question

LEt $A,B \subseteq \mathbb{R}^d$ be non-empty sets. Define their distance to be

$$ d(A,B) = \inf \{ ||x-y|| : x \in A, \; \; y \in B \} $$

For any $A,B$, do we have that $d(A,B) = d( \overline{A}, \overline{B} ) $ ??