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