Exactly the same argument works for Möbius transformations on R and the open semigroup taking the closed unit sphere || x || ≤ 1 into the open unit sphere || x || < 1. The closure is a maximal proper semigroup in the group of all Möbius transformations. When n = 1, the closure corresponds to Möbius transformations of the real line taking the closed interval [–1,1] into itself.