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.]
 A: [EDIT: This answer refers to an older version of the question, which asked for a necessary and sufficient condition on a topological space $X$ for it to satisfy the following axiom: For any two subsets $A$ and $B$ of $X$, we have the logical equivalence $A\subseteq B^{\circ}\Longleftrightarrow A^{-}\subseteq B$.]
Your $A\subseteq B^{\circ}\Longleftrightarrow A^{-}\subseteq B$ axiom holds if
and only if the closed sets of $X$ are precisely the open sets of $X$.
Proof. $\Longrightarrow:$ Assume that the $A\subseteq B^{\circ
}\Longleftrightarrow A^{-}\subseteq B$ axiom holds. Let $S$ be a closed set of
$X$. Then, $S^{-}=S\subseteq S$. Now, applying the $A\subseteq B^{\circ
}\Longleftrightarrow A^{-}\subseteq B$ axiom to $A=S$ and $B=S$ shows that
$S\subseteq S^{\circ}\Longleftrightarrow S^{-}\subseteq S$. Since
$S^{-}\subseteq S$ holds, we thus obtain $S\subseteq S^{\circ}$. Combined with
$S^{\circ}\subseteq S$, this yields that $S=S^{\circ}$, and thus $S$ is open
(since $S^{\circ}$ is open). Thus, we have shown that every closed set $S$ of
$X$ is open. In other words, all closed sets of $X$ are open. Applying this to
the complement set, we conclude that all open sets of $X$ are closed. Thus,
the closed sets of $X$ are precisely the open sets of $X$. This proves the
$\Longrightarrow$ direction of my claim.
$\Longleftarrow:$ Assume that the closed sets of $X$ are precisely the open
sets of $X$. We need to show that the $A\subseteq B^{\circ}\Longleftrightarrow
A^{-}\subseteq B$ axiom holds.
Let $A$ and $B$ be two subsets of $X$. Then, the set $B^{\circ}$ is open and
thus closed (since the closed sets of $X$ are precisely the open sets of $X$);
hence, $\left(  B^{\circ}\right)  ^{-}=B^{\circ}$. Similarly, $\left(
A^{-}\right)  ^{\circ}=A^{-}$. Now, if $A\subseteq B^{\circ}$, then
$A^{-}\subseteq\left(  B^{\circ}\right)  ^{-}=B^{\circ}\subseteq B$.
Conversely, if $A^{-}\subseteq B$, then $A\subseteq A^{-}=\left(
A^{-}\right)  ^{\circ}\subseteq B^{\circ}$ (since $A^{-}\subseteq B$). Hence,
$A\subseteq B^{\circ}\Longleftrightarrow A^{-}\subseteq B$. This proves the
$\Longleftarrow$ direction of my claim.
Thanks to Stefan Hamcke and Martti Karvonen for correcting my proof!
A: Put $A=B=S$, where $S$ is any subset of $X$. Then
$$
\overline S\subseteq S \iff S\subseteq\mathring S
$$
hence
$$
S \text{ is closed} \iff S \text{ is open}
$$
Conversely, assume that every open set is closed (which implies that any closed set is open). If $A$ and $B$ are subsets of $X$ such that $\overline A ⊆ B$, then we also have $A ⊆ \overline A ⊆ \mathring B$, and if $A ⊆ \mathring B$, then $\overline A ⊆ \mathring B ⊆ B$. Hence such a space satisfies
$$
\overline A ⊆ B \iff A \subseteq \mathring B
$$
A: Apparently the elements of $P$ are called regular open sets and the elements of $Q$ are called regular closed sets. Interestingly, the lattices $P$ and $Q$ are Boolean with the function $A^\bot:=X-A^-$ acting as a complement on $P$ and the function $B^\top:=X-B^\circ$ acting as a complement on $Q$.
http://planetmath.org/regularopenset
I didn't find a full characterization of these sets, but apparently every convex open set in $\mathbb{R}^n$ is regular:
Is convex open set in $\mathbb{R}^n$ is regular?
A: 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.)
