A Möbius band of constant positive curvature cannot be complete, since it is known that the only complete surfaces of constant positive curvature are the sphere and the projective plane. The projective plane P2 of constant curvature +1 may be constructed as the quotient of the unit sphere "S"2 in R3 by the antipodal map "A": "S"2 → "S"2, defined by . The open Möbius band is homeomorphic to the once-punctured projective plane, that is, P2 with any one point removed. This may be thought of as the closest that a Möbius band of constant positive curvature can get to being a complete surface: just one point away.