Editing Real projective space (section)

===Construction===
As with all [[projective space]]s, {{tmath|\mathbb{RP}^n}} is formed by taking the [[Quotient space (topology)|quotient]] of <math>\R^{n+1}\setminus \{0\}</math> under the [[equivalence relation]] {{tmath|x\sim \lambda x}} for all [[real number]]s {{tmath|\lambda\neq 0}}. For all {{tmath|x}} in <math>\R^{n+1}\setminus \{0\}</math> one can always find a {{tmath|\lambda}} such that {{tmath|\lambda x}} has [[Norm (mathematics)|norm]]&nbsp;1. There are precisely two such {{tmath|\lambda}} differing by sign. Thus {{tmath|\mathbb{RP}^n}} can also be formed by identifying [[antipodal point]]s of the unit {{tmath|n}}-[[sphere]], {{tmath|S^n}}, in <math>\R^{n+1}</math>.

One can further restrict to the upper hemisphere of {{tmath|S^n}} and merely identify antipodal points on the bounding equator. This shows that {{tmath|\mathbb{RP}^n}} is also equivalent to the closed {{tmath|n}}-dimensional disk, {{tmath|D^n}}, with antipodal points on the boundary, <math>\partial D^n=S^{n-1}</math>, identified.