TITLE: Is every open set the interior of a closed set?
QUESTION [30 upvotes]: I am wondering if this is generally true for any topology. I think there might be counter examples, but I am having trouble generating them.

REPLY [17 votes]: Since the complement of an open set is closed (and vice versa), and since the complement of the interior is the closure of the complement, we can rephrase your question equivalently as:

Is every closed set the closure of some open set?

This immediately suggests a counterexample: any singleton (i.e. a set containing only one point) is closed in $\mathbb R^n$ (with the usual Euclidean topology), but has no non-empty open subsets that it could be the closure of.
Conversely, the complement of any singleton (i.e. $\mathbb R^n \setminus \{x\}$ for any $x \in \mathbb R^n$) provides a counterexample to your original claim, being an open set that cannot be the interior of any closed set.