Call a "blot" set, which is the closure of its interior, the boundary is locally connected, and when you remove boundary blot remains connected. Suppose that there is a blot on the surface of the ndimensional sphere. Is it true that every homeomorphism of the blots on itself extends to a homeomorphism of the sphere on itself? If this is not true, can it be imposed on the blot any additional reasonable conditions (eg, smoothness) to the statement was true?
