Editing Projective space (section)

== Topology ==
{{hatnote|In this section, all projective spaces are real projective spaces of finite dimension. However everything applies to complex projective spaces, with slight modifications.}}

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

Let {{mvar|S}} be the [[unit sphere]] in a normed vector space {{mvar|V}}, and consider the function
<math display="block">\pi: S \to \mathbf P(V)</math>
that maps a point of {{mvar|S}} to the vector line passing through it. This function is continuous and surjective. The inverse image of every point of {{math|'''P'''(''V'')}} consist of two [[antipodal point]]s. As spheres are [[compact space]]s, it follows that: 
{{block indent | em = 1.5 | text =''A (finite dimensional) projective space is compact''.}}

For every point {{mvar|P}} of {{mvar|S}}, the restriction of {{pi}} to a neighborhood of {{mvar|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 (topology)|atlas]] can be provided, as follows.

As soon as a basis has been chosen for {{mvar|V}}, any vector can be identified with its coordinates on the basis, and any point of {{math|'''P'''(''V'')}} may be identified with its [[homogeneous coordinates]]. For {{math|1=''i'' = 0, ..., ''n''}}, the set
<math display="block">U_i = \{[x_0:\cdots: x_n], x_i \neq 0\}</math>
is an open subset of {{math|'''P'''(''V'')}}, and 
<math display="block">\mathbf P(V) = \bigcup_{i=0}^n U_i</math>
since every point of {{math|'''P'''(''V'')}} has at least one nonzero coordinate.

To each {{math|''U''{{sub|''i''}}}} is associated a [[chart (topology)|chart]], which is the [[homeomorphism]]s
<math display="block">\begin{align}
\mathbb \varphi_i: R^n &\to U_i\\
(y_0,\dots,\widehat{y_i},\dots, y_n)&\mapsto [y_0:\cdots:y_{i-1}:1:y_{i+1}:\cdots:y_n],
\end{align}</math>
such that 
<math display="block">\varphi_i^{-1}\left([x_0:\cdots:x_n]\right) =\left (\frac{x_0}{x_i}, \dots, \widehat{\frac{x_i}{x_i}}, \dots, \frac{x_n}{x_i} \right ),</math>
where hats means that the corresponding term is missing.
[[File:P1 ako varieta.png|thumb|200px|right|Manifold structure of the real projective line]]

These charts form an [[atlas (topology)|atlas]], and, as the [[transition map]]s are [[analytic function]]s, it results that projective spaces are [[analytic manifold]]s.

For example, in the case of {{math|1=''n'' = 1}}, that is of a projective line, there are only two {{math|''U''{{sub|''i''}}}}, 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 
<math display="block">x\mapsto \frac 1 x</math>
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.

=== CW complex structure ===
Real projective spaces have a simple [[CW complex]] structure, as {{math|'''P'''<sup>''n''</sup>('''R''')}} can be obtained from {{math|'''P'''<sup>''n''−1</sup>('''R''')}} by attaching an {{math|''n''}}-cell with the quotient projection {{math|'''S'''<sup>''n''−1</sup> → '''P'''<sup>''n''−1</sup>('''R''')}} as the attaching map.