Simple question on a quotient space of real projective space A silly little detail I am unable to work out on my own that is needed for a bigger thing...
I am quite convinced that the quotient space $\mathbb{R} \text{P}^{n} / \mathbb{R} \text{P}^{n-1}$ (where $\mathbb{R} \text{P}^{n-1}$ is included in $\mathbb{R} \text{P}^{n}$ in the standard way) is homeomorphic to $S^n$ on the grounds that it "makes sense" thinking about it geometrically in the case of $n=2$. Nevertheless, I am unable to actually construct a proof of the proposition.
If it is true, then this should be a standard result. Alas I cannot find it looking around. Anyone able to help me?
 A: $\newcommand{\Reals}{\mathbf{R}}\newcommand{\Proj}{\mathbf{P}}\newcommand{\Vecx}{\mathbf{x}}$Consider the continuous mapping $\Vecx \mapsto (\Vecx, \sqrt{1 - \|\Vecx\|^{2}}$ from the closed unit ball in $\Reals^{n}$ to the "upper hemisphere" $H^{+}$ of the unit sphere in $\Reals^{n+1}$. Projection from $\Reals^{n+1}$ minus the origin sends $H^{+}$ onto $\Reals\Proj^{n}$, identifying antipodal points on the boundary. The mapping induced on the unit ball by quotienting out the projective hyperplane $\Reals\Proj^{n-1}$ collapses the boundary sphere to a point. This composite mapping is a continuous bijection on a compact Hausdorff space, so is a homeomorphism.
A: Real projective $n$-space can also be realized as a unit $n$-sphere mod $\pm 1$. Define a map $\sigma$ on the unit $n$-sphere by $$\sigma(x_0, \ldots, x_n) = (x_0^2, x_0 x_1, \ldots, x_0 x_n).$$ Then $\sigma$ is $\pm 1$ invariant and therefore well defined on projective $n$-space. As a projective map, $\sigma$ is injective on the complement of the projective $(n-1)$-space $x_0=0$, which it maps to $0$.  The image of $\sigma$ is the sphere of radius $\tfrac12$ centered at $(\tfrac12, 0, \ldots, 0)$.  So $\sigma$ identifies the quotient with a sphere.
