# Prove that $Y$ is the closure of some open set if and only if $Y$ is the closure of its interior.

Prove that $$Y$$ is the closure of some open set if and only if $$Y$$ is the closure of its interior.

$$\implies:$$ Let $$Y=\overline Z$$ for some open set $$Z$$. Then $$Y$$ is the smallest closed set containing $$Z$$. Since $$\mathring Y$$ is the largest open set contained in $$Y$$, therefore $$Z\subseteq \mathring Y$$. However, since $$\overline{\mathring Y}$$ is the smallest closed set containing $$\mathring Y$$, we have that $$\overline{\mathring Y} \subseteq Y$$, but being that $$Y$$ is the smallest closed set containing $$Z$$ it is also the case that $$Y\subseteq \overline{\mathring Y}$$ and hence $$Y= \overline{\mathring Y}$$.

$$\Longleftarrow:$$ Assume that $$Y=\overline{\mathring Y}$$. Since $$\mathring Y$$ is the largest open set contained in $$Y$$, for any other open set $$Z$$ in $$Y$$ we must have that $$Z\subseteq \mathring Y$$. And since $$\overline Z$$ is the smallest closed set containing $$Z$$, we have $$\overline Z \subseteq \overline{\mathring Y}$$. However, since $$\overline{\mathring Y}$$ is the smallest closed set containing $$\mathring Y$$, it is also true that $$\overline{\mathring Y} \subseteq \overline Z$$, so $$\overline Z = \overline{\mathring Y}=Y$$. $$\Box$$

Is this correct, and can this be simplified at all? The two directions seem largely similar in reasoning.

This problem confused me for a while since it was basically equivalent to saying, "Y is the smallest closed set containing an open set Z if and only if Y is the smallest closed set containing the largest open set contained in Y" which was hard for me to parse at first.

The proof of the forward direction is fine. After showing $$Z\subseteq \mathring Y$$ in the first paragraph, you can alternatively conclude by writing $$Y = \overline{Z} \subseteq \overline{\mathring Y} \subseteq \overline{Y} = Y$$ which proves $$\overline{\mathring Y} = Y$$. In the second paragraph, it would have been enough to note that since $$\mathring Y$$ is open, the reverse direction is proved.