Projective space
Topology

A projective space is a topological space, as endowed with the quotient topology of the topology of a finite dimensional real vector space.

Let S be the unit sphere in a normed vector space V, and consider the function π: S → P (V) {\displaystyle \pi:S\to \mathbf {P} (V)} that maps a point of S to the vector line passing through it. This function is continuous and surjective. The inverse image of every point of P (V) consist of two antipodal points. As spheres are compact spaces, it follows that:

For every point P of S, the restriction of π to a neighborhood of P is a homeomorphism onto its image, provided that the neighborhood is small enough for not containing any pair of antipodal points. This shows that a projective space is a manifold. A simple atlas can be provided, as follows.

As soon as a basis has been chosen for V, any vector can be identified with its coordinates on the basis, and any point of P (V) may be identified with its homogeneous coordinates. For i = 0,..., n, the set U i = {, x i ≠ 0 } {\displaystyle U_{i}=\{,x_{i}\neq 0\}} is an open subset of P (V), and P (V) = ⋃ i = 0 n U i {\displaystyle \mathbf {P} (V)=\bigcup _{i=0}^{n}U_{i}} since every point of P (V) has at least one nonzero coordinate.

To each U is associated a chart, which is the homeomorphisms φ i: R n → U i (y 0, …, y i ^, … y n) ↦, {\displaystyle {\begin{aligned}\mathbb {\varphi } _{i}:R^{n}&\to U_{i}\\(y_{0},\dots,{\widehat {y_{i}}},\dots y_{n})&\mapsto,\end{aligned}}} such that φ i − 1 () = (x 0 x i, …, x i x i ^, …, x n x i), {\displaystyle \varphi _{i}^{-1}\left(\right)=\left({\frac {x_{0}}{x_{i}}},\dots,{\widehat {\frac {x_{i}}{x_{i}}}},\dots,{\frac {x_{n}}{x_{i}}}\right),} where hats means that the corresponding term is missing.

These charts form an atlas, and, as the transition maps are analytic functions, it results that projective spaces are analytic manifolds.

For example, in the case of n = 1, that is of a projective line, there are only two U, which can each be identified to a copy of the real line. In both lines, the intersection of the two charts is the set of nonzero real numbers, and the transition map is x ↦ 1 x {\displaystyle x\mapsto {\frac {1}{x}}} in both directions. The image represents the projective line as a circle where antipodal points are identified, and shows the two homeomorphisms of a real line to the projective line; as antipodal points are identified, the image of each line is represented as an open half circle, which can be identified with the projective line with a single point removed.
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