In mathematics, real projective space, denoted \mathbb{RP}^n or \mathbb{P}_n(\R), is the topological space of lines passing through the origin 0 in the real space \R^{n+1}. It is a compact, smooth manifold of dimension n, and is a special case \mathbf{Gr}(1, \R^{n+1}) of a Grassmannian space.Basic properties
Construction
As with all projective spaces, RPn is formed by taking the quotient of Rn+1 ∖ under the equivalence relation x ∼ λx for all real numbers λ ≠ 0. For all x in Rn+1 ∖ one can always find a λ such that λx has norm 1. There are precisely two such λ differing by sign.Thus RPn can also be formed by identifying antipodal points of the unit n-sphere, Sn, in Rn+1.One can further restrict to the upper hemisphere of Sn and merely identify antipodal points on the bounding equator. This shows that RPn is also equivalent to the closed n-dimensional disk, Dn, with antipodal points on the boundary, ∂Dn = Sn−1, identified.Low-dimension
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