Let $n \geq 0$ be an integer. Define

\begin{align*} S^n := \{x \in \mathbb{R}^{n+1} : |x| = 1\}. \end{align*}

Let $\mathbb{R}^{\times} := \mathbb{R}\setminus\{0\}$. Let $\mathbb{R}\mathbb{P}^n$ denote the quotient [topological space](/page/Topological%20Space) of $\mathbb{R}^{n+1}\setminus\{0\}$ by the [equivalence relation](/page/Equivalence%20Relation) $v \approx w$ if and only if $w = \lambda v$ for some $\lambda \in \mathbb{R}^{\times}$. Write the quotient map as

\begin{align*} p: \mathbb{R}^{n+1}\setminus\{0\} \to \mathbb{R}\mathbb{P}^n. \end{align*}

For $v \in \mathbb{R}^{n+1}\setminus\{0\}$, write $p(v) = \mathbb{R}v$. Define an equivalence relation $\sim$ on $S^n$ by $x \sim y$ if and only if $y = x$ or $y = -x$. Then there is a canonical homeomorphism

\begin{align*} S^n/{\sim} \cong \mathbb{R}\mathbb{P}^n \end{align*}

given by sending the equivalence class of $x \in S^n$ to the projective point represented by the line $\mathbb{R}x \subset \mathbb{R}^{n+1}$.