Real projective space

In mathematics, real projective space, denoted ⁠${\displaystyle \mathbb {RP} ^{n}}$⁠ or ⁠${\displaystyle \mathbb {P} _{n}(\mathbb {R} ),}$⁠ is the topological space of lines passing through the origin 0 in the real space ⁠${\displaystyle \mathbb {R} ^{n+1}.}$⁠ It is a compact, smooth manifold of dimension n, and is a special case ⁠${\displaystyle \mathbf {Gr} (1,\mathbb {R} ^{n+1})}$⁠ of a Grassmannian space.