Consider the unit sphere $S^n$ with the equivalence relation that identifies antipodal points of $S^n$. Show that the quotient space $S^n/\sim$ is homeomorphic to the projective space $\mathbb{RP}^n$.  I know that $\mathbb{RP}^n=\overline{B^n}/\sim$, so if I can define a homeomorphism from $\overline{B^n}$ to $S^n$, I think that should be sufficient. My thought was to define a function on the ball so that every point on a line that goes through the origin is mapped to the outermost point, a point on the sphere. But I don't know how to write this explicitly. I have seen questions like this stated, with the explanation I came up with given, but there is no explicit definition of the homeomorphism.
I saw this in Basic Topology by M.A.Armstrong. It gives three descriptions of real n-dimensional  projective space $P^n$. Two of them are:  (a) Begin with the unit sphere $S^n$ in $E^{n+1}$ and identify its antipodal points.  (c) Begin with the unit ball $B^n$ and identify antipodal points of its boundary sphere.  I find it hard to imagine why these two descriptions lead to the same space.  Can you please help? Thank you.
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