High Quality Content by WIKIPEDIA articles! In mathematics, real projective space, or RPn is the projective space of lines in Rn+1. It is a compact, smooth manifold of dimension n, and a special case of a Grassmannian. As with all projective spaces, RPn is formed by taking the quotient of Rn+1 ? {0} under the equivalence relation x ? ?x for all real numbers ? ? 0. For all x in Rn+1 ? {0} one can always find a ? such that ?x has norm 1. There are precisely two such ? 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.

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