Prove that $A\cap \partial D\neq\varnothing$ under some conditions

Question

Let $A, B$ and $D$ be non-empty subsets of $\mathbb{C}$.

Assume that

• $A\cap \partial B\neq\varnothing$

• $B\subset D$.

• $\overline{B}=\overline{D}$.

Is $A\cap \partial D\neq\varnothing$?

Answer 1

NO. Take $B=\mathbb Q, A=D=\mathbb R$ for a counter-example.