As with all projective spaces, RPn is formed by taking the quotient of Rn+1 − {0} under the equivalence relationx ∼ 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 bounday, ∂Dn = Sn−1, identified.
The antipodal map on the n-sphere (the map sending x to −x) generates a Z2group 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.
Projectivespaces are essential to algebraic geometry through the rich field of projective geometry developed in the nineteenth century, but also in the constructions of the modern theory (based on graded algebras).
Projectivespaces and their generalisation to flag manifolds also play a big part in topology, the theory of Lie groups and algebraic groups, and their representation theory.
The use of projectivespaces makes quite rigorous the talk about a 'line at infinity' (where parallel lines meet), or a 'plane at infinity' for three dimensions: a translation of the latter can be made as part of the projectivespace associated to a four-dimensional real vector space.
In mathematics, a projective plane has two possible definitions, one of them coming from linear algebra, and another (which is more general) coming from the combinatorics of block designs.
The most common projective plane is the realprojective plane, which is a topological surface with surprising geometric properties; after that is the complex projective plane of algebraic geometry, a topological four-dimensional manifold.
In the case of finite projective planes, the only proof known of the purely geometric statement that Desargues theorem then implies Pappus' theorem (the converse being always true and provable geometrically) is through this algebraic route, using Wedderburn's theorem that finite division rings must be commutative.
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