Sets $A, B$ disjoint $\implies \overline{A} \cap B = A \cap \overline{B} = \emptyset$?

Question

If a set $X \subset \mathbb{R}^n$ is disconnected/separated, then it can be written as the union of two disjoint, relatively open subsets, say $A$ and $B$; that is $$X = A \cup B.$$ I keep reading that the above statement implies the following about the closures of $A$ and $B$, written as $\overline{A}$ and $\overline{B}$: $$\overline{A} \cap B = A \cap \overline{B} = \emptyset.$$ I don't understand why the fact that $A$ and $B$ are disjoint would imply that each of $A$ and $B$ are also disjoint with the closure of the other.

Answer 1

$B$ is open, therefore $X\setminus B$ is closed and $A\subset X\setminus B$, therefore $\overline{A}\subset X\setminus B$, i.e., $\overline{A}\cap B=\emptyset$.

So this actually shows the result without using that their union is $X$.

Answer 2

Well, it assumes the condition that $A$ and $B$ are disjoint open sets with $A\cup B=X$.
Then, $X\setminus A=B$ is open, hence $A$ is closed: $\bar A=A$, and the claim follows.
(Note, here $\bar A$ means the closure within the subspace $X$.)