> 
> Let $X$ and $Y$ be subsets of a metric space. If $int\overline{X} = int\overline{Y} = \emptyset$ then $int(\overline{X\cup Y}) = \emptyset$.
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I know that $int(\overline{X\cup Y}) = int(\overline{X}\cup \overline{Y})$ and so I'm trying to prove the more general statement 
$$
int(X) = int(Y) = \emptyset \implies int(X\cup Y) = \emptyset
$$
which seems true to me but I not getting the proof. 

What I tried so far is to consider an $x\in int(X\cup Y)$, so there is a open ball $B\_x$ that is contained in $X\cup Y$ but it cannot be contained in $X$ or $Y$ because these have empty interior. But I don't know what to do next. 

Thanks in advance for any help.