Wikipedia describes Tarski's circle-squaring problem like this:
Tarski's circle-squaring problem is the challenge, posed by Alfred Tarski in 1925, to take a disc in the plane, cut it into finitely many pieces, and reassemble the pieces so as to get a square of equal area.
To get to my question, I need to discuss a little bit about two very different types of dissection problem. (Note: The notation $\cong$ means congruence.)
Now, from my understanding, most dissection puzzles can be formalized as in the following way: Let $A$ and $B$ be regular open subsets of the plane. Can you write $\overline A=\bigcup\limits_{i=1}^n\overline{A_i}$ and $\overline B=\bigcup\limits_{i=1}^n\overline{B_i}$ such that the $A_i$ (resp. $B_i$) are disjoint regular open sets, and such that for each $i$, $A_i\cong B_i$? (Regular open sets are those which are the interiors of their closures.)
The benefit of such a formalization is that we do not have to worry about individual points. For example, Dudeney famously provided a dissection of a triangle into a square. (In fact, it's a hinged dissection, as you can see demonstrated here.) Dudeney's dissection easily fits within this paradigm.
However, we can also ask the following sort of dissection puzzle: Let $A$ and $B$ be subsets of the plane. Can you write $A=\bigcup\limits_{i=1}^nA_i$ and $B=\bigcup\limits_{i=1}^nB_i$ such that the $A_i$ (resp. $B_i$) are disjoint sets, and such that for each $i$, $A_i\cong B_i$?
This is a much stricter thing to ask; we now have to care about individual points! Dudeney's dissection does not satisfy this version of dissection at all. (On the other hand, this does describe the dissection in the Banach–Tarski paradox, for example.)
For the sake of giving names to things, let's call the former the classical dissection problem, and the latter the shapes-as-sets dissection problem. (I choose this name for it because believe it was only made possible by the modern shapes-as-sets paradigm, so useful in topology and related fields, in which shapes are modeled as the set of points lying on them, and in which (conversely) there are no restrictions on which sets of points can be considered shapes.)
Now, to get back to Tarski. I see on Wikipedia that Tarski's circle-squaring problem has been solved in the affirmative: it is possible to decompose a disc into finitely many pieces and rearrange them into a square. I would like to ask for clarification: is it the classical circle-squaring problem or the shapes-as-sets circle-squaring problem that has been solved (in the senses described above?
Furthermore, on the assumption that it is the latter: is the classical circle-squaring problem (as formalized above in terms of regular open sets and their closures) possible? My instinct is no, but I'm not sure how to prove it. I see Wikipedia has a reference to an impossibility result when the pieces are cut by Jordan curves, but it's not immediately obvious to me that this is equivalent.
EDIT: I'm realizing belatedly that the classical problem could be formalized equivalently and more simply in terms of regular closed sets (that is, closed sets which are the closures of their interiors) whose interiors are disjoint.
EDIT2: I just realized, these two are the same definition relativized to different Boolean algebras. The former is the Boolean algebra of regular open sets and the latter is the power set Boolean algebra; these are both complete Boolean algebras.