If two sets $A,B$ are (topologically) separated, is $m^*(A\cup B)=m^*(A)+m^*(B)$ always true?

Question

By separated, we mean $\overline{A}\cap B=\emptyset$ and $\overline{B}\cap A=\emptyset$. If $dist(A,B)>0$, then we know that $m^*(A\cup B)=m^*(A)+m^*(B)$. If $A\cap B=\emptyset$, we know that there is a counterexample here.

BTW, the outter meausre $m^*$ here is the one for Lebesgue measure.