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? 

So, a "point" in $\mathbb{RP}^n$ is secretly the same thing as a pair of antipodal points. But, if you look at two antipodal points in $S^n$, one of two things occurs:


*

*One of the points lies in the open hemisphere $\mathrm{Int}(D^n)$, and the other is in the opposite open hemisphere.

*Both points are on the equator $\partial D^n$.
So, every point in $\mathbb{RP}^n$ has a representative in $D^n$, and it has in fact only one such representative, except if it lies on the equator. So you get $\mathbb{RP}^n$ by considering $D^n$ and "correcting" the only injectivity default by collapsing the pairs of antipodal points in the equator.
