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? 

Every point on the sphere is mapped to the line in the direction of that point, hence antipodal points are identified as they map to the same line.
Similarly the hemisphere is also surjective on the set of lines via this identification. To make it a bijection (as before) one needs to identify only the antipodal points on the boundary, as the antipodal points to the internal points of the hemisphere lie outside the hemisphere, and so there (on the internal points) the map is already 1:1
