[[Projective space]]
# Real projective space

The $n$-dimensional **real projective space** $\mathrm{P}^n \mathbb{R} = \mathrm{P}(\mathbb{R}^{n+1})$ is a [[Compact space|compact]] $C^\omega$-[[differentiable manifold|manifold]] extending Euclidean space such that parallel lines intersect at infinity.
It is equivalently characterized as #m/def/geo/diff

- The [[Grassmannian]] $\mathrm{Gr}_{1}(\mathbb{R}^{n+1})$, i.e. the space of 1-dimensional subspaces of $\mathbb{R}^{n+1}$
- The [[Quotient topology|quotient]] $(\mathbb{R}^n \setminus \{ 0 \})/\mathbb{R}^\times$, i.e. vectors on the same 1-dimensional subspace are identified
- The $n$-[[Sphere space|sphere]] with antipodal points identified, i.e. a quotient $\mathbb{B}^n /{\sim}$

> [!tip]+ Intuition
> Consider lines in $\mathbb{R}^3$ intersecting the origin.
> By selecting some “projecting plane” above the origin,
> one may label _almost all_ such lines by their unique intersection point.
> What remains are lines parallel to the plane,
> so $\mathrm{P}^2 \mathbb{R} \cong \mathbb{R}^2 \amalg \mathrm{P}^{2}\mathbb{R}$ where the latter component are called the “points at infinity”
> 
> Now in the other direction, a line in the projective plane $\mathrm{P}^2 \mathbb{R}$ corresponds to a plane in $\mathbb{R}^3$.


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