Книга: Real Projective Space: Mathematics, Projective Space, Differentiable Manifold, Grassmannian, Equivalence Relation, Antipodal Point, Covering Space, Simply Connected Space, Curve, Fibration

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.