Assume you've got an arbitrary topological space $X$. Now let $I$ be the set of the interiors of all closed subsets of $X$. And now assume you give me $I$, but don't tell me what $X$ is. Can I reconstruct the topology from $I$ alone? If it is not always possible, does there exist any commonly assumed additional feature of topological spaces so that if I restrict $X$ to topological spaces with that feature, the reconstruction is possible?
Note that while all members of $I$ are open by construction, generally not all open sets of $X$ will be in $I$. For example on $\mathbb R$, the set $(-1,0)\cup(0,1)$ is open, but not the interior of a closed set.