The Galois connection between topological closure and topological interior [Update: I changed the question so that $-$ is only applied to closed sets and $\circ$ is only applied to open sets.]
Let $X$ be a topological space with open sets $\mathcal{O}\subseteq 2^X$ and closed sets $\mathcal{C}\subseteq 2^X$. Consider the pair of maps $-:\mathcal{O}\leftrightarrows\mathcal{C}:\circ$ where $-$ is the topological closure and $\circ$ is the topological interior. Under what conditions will it be true that for all $A\in\mathcal{O}$ and $B\in\mathcal{C}$ we have $$A\subseteq B^\circ \Longleftrightarrow A^-\subseteq B \,\,?$$
[Update: Darij showed that this condition holds for all topological spaces.] If this condition does hold then by general nonsense (the theory of abstract Galois connections), we obtain an isomorphism of lattices $$-:P\approx Q:\circ$$
where $P$ is the the lattice of sets $A\in\mathcal{O}$ such that $(A^-)^\circ=A$ and $Q$ is the lattice of sets $B\in\mathcal{C}$ such that $(B^\circ)^-=B$, where both $P$ and $Q$ are partially ordered by inclusion.
The existence of this lattice isomorphism makes me wonder: is there a nice characterization of the elements of $P$ and $Q$? Certainly not every open set is in $P$. For example, if $X=\mathbb{R}$ with the usual topology then the set $(0,1)\cup (1,2)$ is open, but $$(((0,1)\cup(1,2))^-)^\circ = ([0,2])^\circ = (0,2) \supsetneq (0,1)\cup(1,2).$$
[Update: I found the answer. See below.]

Suppose $B$ is open and $A=B$. The condition implies that $Cl(B)=Cl(A) \subset B$ (where $Cl$ denotes closure). So the condition implies that every open set is closed. It is easily verified that if every open set is closed then the condition is satisfied for all $A,B$. So the condition is equivalent to "Every open set is closed." If the space $X$ is a $T_1$ space (which means that $\{p\}$ is closed for every $ p \in X $) then $X$ is discrete ,as  $X$ \ $\{p \}$ open $\implies $ $ X$ \ $\{p \}$ closed $\implies$ $\{p\}$ open for all $p \in X$. The coarse topology also satisfies the condition, and there are many others in general. For example if $F$ is a pairwise-disjoint family of subsets of $X$ with $ \cup F =X$, let $F$ be a base for a topology.(Extra : I've just noticed that a $T_0$ space that satifies the condition is a $T_1$ space and hence discrete.)
