I've wondered about the following question, whose answer is perhaps well known (in this case I apologize in advance).
The Lakes of Wada are a famous example of three disjoint connected open sets of the plane with the counterintuitive property that they all have the same boundary (!)
My question is the following :
Can we find four disjoint connected open sets of the plane that have the same boundary?
More generally :
For each $n \geq 3$, does there exist $n$ disjoint connected open sets of the plane that have the same boundary? If not, then what is the smallest $n$ such that the answer is no?
Thank you, Malik