$\Delta_2$-inseparability? Since long ago it is known the existence of non-recursive sets (i.e., non-$\Delta_1$), e.g., Halting problem. It is also known (firstly noticed by Trakhtenbrot, and deeply studied by Smullyan) the existence of (non-trivial) pairs of sets that are recursively inseparable; this means that there are sets $A,B \subseteq \mathbb{N}$ such that $A \subsetneq B$, and there is no recursive set $C$ such that $A \subseteq C \subseteq B$. An example of pair with this property is to consider $A$ as the set of (Gödel numbers of) first-order valid formulas, and $B$ as the set of (Gödel numbers of) first-order formulas valid in all finite structures.
I am interested on what it is known about the "natural" generalization of the provious notion to $\Delta_2$sets, and to $\Delta_3$, etc. To be more precise: what is it known about pairs of sets $A,B \subseteq \mathbb{N}$ such that:


*

*$A \subsetneq B$,

*there is no $\Delta_2$ set $C$ such that $A \subseteq C \subseteq B$.


Is there any such pair? Is there a general theory for this notion as the one developed by Smullyan for $\Delta_1$? Has this been studied in some papers?
Update 1: I am interested in non trivial and explicit examples. For the non trivial part it makes more sense to replace $A \subsetneq B$ with the following conditions: $A \subseteq B$, and $B \setminus A$ is infinite.
Update 2: To make my question closer to what happens in the $\Delta_1$ example given above let me add the constraints that $A$ is $\Sigma_2$-complete and B is $\Pi_2$-complete.
 A: Since you put no restrictions on the complexity of $A$ and $B$, this should be easy.  Given any countable family $X$ of subsets of $\mathbb N$, there are sets $A\subseteq B$ with $B-A$ infinite and with no $C\in X$ satisfying $A\subseteq C\subseteq B$.  List the elements of $X$ in an $\omega$-sequence $(C_n)$ and build $A$ and $B$ inductively as follows.  At every stage, you will have decided for only finitely many natural numbers $k$ whether $k\in A$, $k\in B-A$, or $k\notin B$.  At stage $n$, if $C_n$ has an element about which no decision has been made yet, choose one such element and put it out of $B$, thus ensuring that $C_n$ won't be a subset of $B$.  Otherwise, choose an element outside $C_n$ about which no decision has been made and put that element into $A$, thus ensuring that $A$ won't be a subset of $C_n$.  Finally (in stage $n$), choose another element about which no decision has yet been made and put it into $B-A$, thus ensuring that, at the end of the construction, $B-A$ will be infinite.
If $X$ is a class like $\Delta^0_2$ and you use a "reasonable" enumeration of it, and if, wherever I said "choose", you always take the smallest available number, then this process will give you some computability information about $A$ and $B$.  You might, however, be able to do better by being more subtle in the construction.
A: It may be worth noting that the example of recursively inseparable sets (a la boumol's definition) in the question is very much related to a standard example of recursively inseparable sets (a la the traditional definition): Let $A := \{ e : \varphi_e(e) \downarrow = 0 \}$ and $B := \{ e : \varphi_e(e) \downarrow = 1 \}$.  Then $(A,B)$ is recursively inseparable (a la the traditional definition).
Indeed, direct relativization of this result to the Halting Problem $K$ offers an explicit example of a pair of $\Delta^0_2$-inseparable sets.  
CLAIM: Let $A := \{ e : \varphi^K_e(e) \downarrow = 0 \}$ and $B := \{ e : \varphi^K_e(e) \downarrow = 1 \}$.  Then $(A,B)$ is $\Delta^0_2$-inseparable (a la the traditional definition).  
PROOF: Towards a contradiction, fix a $\Delta^0_2$ set $C$ separating $A$ and $B$.  Let $z$ be an index so that $C = \varphi^K_z$.  Being a characteristic function, the function $\varphi^K_z$ satisfies either $\varphi^K_z(z) \downarrow = 1$ or $\varphi^K_z(z) \downarrow = 0$.  If the former, then $z \in B$ (by definition) and $z \in C$ (as $C = \varphi^K_z$), contradicting $B \cap C = \emptyset$.  If the latter, then $z \in A$ (by definition) so $z \in C$ (as $A \subseteq C$), contradicting $z \not \in C$ (as $C = \varphi^K_z$).  Hence, in either case, we have a contradiction.
More generally, I expect that many of the results on recursive inseparability relativize in straightforward manners to yield results on, for example, $\Delta^0_2$-inseparability.
