Call a "blot" set, which is the closure of its interior, connected, and when you remove boundary remains connected. Suppose that there is a blot on the surface of the n-dimensional 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, local connectedness of the boundary) to the statement was true?