An abstract 2-dimensional, non-orientable (one-sided) surface without boundary. Like the Klein bottle, there is no way to physically construct (embed) the real projective plane in three dimensions, so models intersect or pass through themselves. However, 4-dimensional models exist without self-intersection.

One definition of the real projective plane is the space of all lines through the origin in 3-dimensional Euclidean space. Topologically, this can be defined simply as the direction vectors for the unit sphere. Then, choosing a hemisphere, since each vector in the other hemisphere defines the same line as one in the chosen hemisphere, the identification of opposite points provides an equivalence, whose quotient space is the real projective plane. See projective geometry, projective space.