Definition:Projective Space

Given a sphere $$\mathbb{S}^n \in \R^{n+1}, \ $$, define an equivalence relation on the sphere by

$$x,y \in \R^{n+1}: x \sim y \iff x = -y \ $$

The real projective space of dimension $$ n \ $$ is the quotient space $$\mathbb{S}^n / \sim \ $$ and is denoted $$\mathbb{RP}^n \ $$.