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.