If two sets have empty interior closures, does the interior of the closures's union is contained in the union of the interior of the closures?

Question

Is that true that $(\overline{X})^{\circ} = (\overline{Y})^{\circ} = \emptyset \Rightarrow (\overline{X \cup Y})^{\circ} \subseteq (\overline{X})^{\circ} \cup (\overline{Y})^{\circ}$, where $^{\circ}$ stands for "interior of the set"?

Answer 1

It's true.

Assume that $U \subseteq \left(\overline{X \cup Y}\right)^\circ$ is open and non-empty. Then $U \setminus \overline{Y}$ is open (as intersection of open $U$ and open complement to $\overline{Y}$) and non-empty (otherwise $U \subseteq \overline{Y}$ and $\overline Y$ has non-empty interior). But $U \setminus \overline{Y} \subseteq \overline{X}$, so $\overline X$ has non-empty interior.