The \(n\)-dimensional real projective space \(\mathbb{R}\mathbb{P}^n\) consists of all one-dimensional subspaces of \(\mathbb{R}^{n+1}\). It is a quotient space induced by \(q\colon \mathbb{R}^{n+1}\setminus \{0\}\to \mathbb{R}\mathbb{P}^n\).
Remarks
- The real projective space \(\mathbb{R}\mathbb{P}^n\) is exactly the orbit space of the action of \(\mathbb{R}\setminus \{0\}\) on \(\mathbb{R}^n\setminus \{0\}\) .
- \(\mathbb{R}\mathbb{P}^n\) is a \(n\)-manifold.
- The real projective space \( \mathbb{RP}^n \) is compact, since it is homeomorphic to \( \mathbb{S}^n \).