let "x" be an "n"-dimensional "topological sphere" in the ("n"+1)-dimensional euclidean space r ("n" > 0), i.e. the image of an injective continuous mapping of the "n"-sphere "s" into r. then the complement "y" of "x" in r consists of exactly two connected components. one of these components is bounded (the interior) and the other is unbounded (the exterior). the set "x" is their common boundary.