Real Projective $n$ space $\mathbb{R}P^{n}$ In example 0.4 of Hatcher, he says that $\mathbb{R}P^{n}$ is just the quotient space of the sphere $S^{n}$ with antipodal points identified. He then says that this is equivalent to the quotient of a hemisphere $D^{n}$ with the antipodal points of the boundary identified. 
I don't understand why those spaces are equivalent. Could someone please explain? 

To be very rigorous: view $D^n$ as an hemisphere included in $S^n$. The antipodal equivalence relation $\sim$ on $S^n$ restricts to the antipodal relation on $D^n$ (so we'll write $\sim$ for both). This means that the inclusion map $D^n \to S^n$ descends to a well-defined map $f:D^n / {\sim} \to S^n / {\sim}$ on the quotient spaces, and it is clear that $f$ is a bijection. By the definition of the quotient topologies on these spaces, $f$ is continuous. But, as a quotient of a compact space, $D^n/{\sim}$ is compact, and it is a good exercise to check by hand that $\mathbb{R}P^n = S^n/{\sim}$ is Hausdorff. Therefore $f$ is closed, continuous and bijective, hence a homeomorphism.
