Why is $D^n/\sim$ homeomorphic to $\mathbb{RP}^n$?
Take these local coordinates, to proof ${D^{n + 1}} \simeq P{\mathbb{R}^n}$ \begin{gathered} {U_i} = \{ \left. {[{x_0}, \cdots ,{x_n}]} \right|{x_i} \ne 0\} \subset P({\mathbb{R}^{n + 1}}) \hfill \\ {\varphi _i}:{U_i} \to {\mathbb{R}^n},[{x_0}, \cdots ,{x_n}] \to (\frac{{{x_0}}}{{{x_i}}}, \cdots ,\frac{{{x_{i - 1}}}}{{{x_i}}},\frac{{{x_{i + 1}}}}{{{x_i}}}, \cdots ,\frac{{{x_n}}}{{{x_i}}}) \hfill \\ {\psi _i}:{\mathbb{R}^n} \to {U_i},({a_1}, \cdots ,{a_n}) \to [{a_1}, \cdots ,{a_{i - 1}},1,{a_i}, \cdots ,{a_n}] \hfill \\ \end{gathered}