A characterization of open sets 
Let $(M,d)$ be a metric space. Then a set $A\subset M$ is open if, and only if, $A \cap \overline X \subset \overline {A \cap X}$ for every $X\subset M$.

This is a problem from metric spaces, but I think that it only requires topology. I don't know how to do it.

Suppose first that $A$ is open. Let $X\subseteq M$, and suppose that $x\in A\cap\operatorname{cl}X$. Let $U$ be any open set containing $x$; then $A\cap U$ is an open neighborhood of $x$, and $x\in\operatorname{cl}X$, so $U\cap X\ne\varnothing$. Thus, $U\cap(A\cap X)\ne\varnothing$, and since $U$ was an arbitrary open set containing $x$, we conclude that $x\in\operatorname{cl}(A\cap X)$. Finally, $x$ was an arbitrary element of $A\cap\operatorname{cl}X$, so $A\cap\operatorname{cl}\subseteq\operatorname{cl}(A\cap X)$. This proves the ($\Rightarrow$) direction.
Here's a hint for the ($\Leftarrow$) direction: suppose that $A$ is not open, let $X=M\setminus A$, and show that $A\cap\operatorname{cl}X\nsubseteq\operatorname{cl}(A\cap X)$. (This is pretty easy to do.)
