Real projective space

Description

Let $n \ge 1$ be an integer. The real projective space $\mathbb{RP}^n$ is the set of lines through the origin in $\R^{n+1}$. As a topological space it is the quotient of the $n$-sphere $S^n$ by the relation that identifies each point $x$ with its antipode $-x$. It is a compact topological manifold of dimension $n$. For $i = 1, \ldots, n+1$, let $U_i$ be the image of $\set{x \in S^n : x_i \ne 0}$. The map $U_i \to \R^n$ sending the class of $x$ to the $n$ ratios $x_j / x_i$ with $j \ne i$ is a homeomorphism, and the $n + 1$ sets $U_i$ cover $\mathbb{RP}^n$.