Real projective space
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In mathematics, real projective space, or RPn is the projective space of lines in Rn+1. The case n = 1 gives the real projective line which is topologically equivalent to a circle. The case n = 2 is called the real projective plane, RP2. The space RPn is a compact, smooth manifold of dimension n. It is 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 ∼ 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.
The antipodal map on the n-sphere (the map sending x to −x) generates a Z2 group action on Sn. As mentioned above, the orbit space for this action is RPn. This action is actually a covering space action giving Sn as a double cover of RPn. Since Sn is simply connected for n ≥ 2, it also serves as the universal cover in these cases. It follows that the fundamental group of RPn is Z2. A generator for the fundamental group is the closed curve obtained by projecting any curve connecting antipodal points in Sn down to RPn.
