Lakes of Wada

Example

The Lakes of Wada are three disjoint, connected, open subsets of the plane that all have the same boundary. They can be built from a closed square of land with three lakes of water in it. At step $n$, each lake in turn is extended by a thin canal of open water that winds through the land. After the step, every point of the remaining land is within distance $2^{-n}$ of each lake. The canals never meet each other, and the land left after each step is a nonempty compact set, so some land remains at the end. After all steps, each lake is the union of its stages, so it is open and connected. Every point of the remaining land is a limit of points of each lake, and the lakes are open and disjoint. So the remaining land is the boundary of each of the three lakes.