Let $X$ be the $n$-dimensional unit sphere $S^{n}=left{x in mathbb{R}^{n+1}:|x|=1 ight}$ with antipodal points identified, and let $A$ be the $(n+1)$-dimensional real vector space without

Let $X$ be the $n$-dimensional unit sphere $S^{n}=\left\{x \in \mathbb{R}^{n+1}:|x|=1\right\}$ with antipodal points identified, and let $A$ be the $(n+1)$-dimensional real vector space without the point at the origin: $A=\mathbb{R}^{n+1} \backslash\{0\}$. Define an equivalence relation $\sim$ among elements of $A$, such that $X$ can be identified with the quotient space of $A$ with respect to $\sim$, i.e., $X=A / \sim$.