Schoenflies conjecture

The conjecture stating that the common boundary of two domains in the plane is always decomposable. A space $X$ is called decomposable if it is connected and can be represented as the union of two closed connected subsets different from $X$. This conjecture was stated by A. Schoenflies in 1908 and disproved by L.E.J. Brouwer in 1910, who discovered indecomposable continua, i.e. connected compact sets that cannot be represented as the union of two closed proper connected subsets. For each finite or countable $k\geq2$, $k$ domains in the plane have been constructed with an indecomposable continuum as common boundary.

References

[1]	K. Kuratowski, "Topology" , 2 , Acad. Press (1968) (Translated from French)