Real projective space

In mathematics, real projective space, or RPⁿ or ${\displaystyle \mathbb {P} _{n}(\mathbb {R} )}$, is the topological space of lines passing through the origin 0 in Rⁿ⁺¹. It is a compact, smooth manifold of dimension n, and is a special case Gr(1, Rⁿ⁺¹) of a Grassmannian space.

As with all projective spaces, RPⁿ is formed by taking the quotient of Rⁿ⁺¹\{0} under the equivalence relation x ∼ λx for all real numbers λ ≠ 0. For all x in Rⁿ⁺¹\{0} 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.

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