Let $X$ be a topological space, $Y \subset X$. Is it true that if $Y \subset O$ and $O$ is open, then $\overline Y \subset O$?
If $X$ is a metric space it is true since if $\bar y \in \overline Y \setminus Y$ then $d(\bar y, Y) = 0$. Does the result still holds for arbitrary topological spaces? If not, what is the essential request to make it work?