Definition:Projective Space

Definition

Let $\Bbb S^n \subseteq \R^{n+1}$ be an $n$-sphere.

Let $\sim$ be the equivalence relation defined on $\Bbb S^n$ by:

$x, y \in \Bbb S^n: x \sim y \iff x = -y$

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