Editing Projective plane (section)

===Classical examples===

The [[real projective plane]] '''RP'''<sup>2</sup> arises when ''K'' is taken to be the [[real number]]s, '''R'''. As a closed, non-orientable real 2-[[manifold]], it serves as a fundamental example in topology.<ref name="Bredon1993">The real projective plane appears 37 times in the index of Bredon (1993), for example.</ref>

In this construction, consider the unit sphere centered at the origin in '''R'''<sup>3</sup>. Each of the '''R'''<sup>3</sup> lines in this construction intersects the sphere at two antipodal points. Since the '''R'''<sup>3</sup> line represents a point of '''RP'''<sup>2</sup>, we will obtain the same model of '''RP'''<sup>2</sup> by identifying the antipodal points of the sphere. The lines of '''RP'''<sup>2</sup> will be the great circles of the sphere after this identification of antipodal points. This description gives the standard model of [[elliptic geometry]].

The [[complex projective plane]] '''CP'''<sup>2</sup> arises when ''K'' is taken to be the [[complex number]]s, '''C'''. It is a closed complex 2-manifold, and hence a closed, orientable real 4-manifold. It and projective planes over other [[Field (algebra)|fields]] (known as '''[[pappian plane]]s''') serve as fundamental examples in [[algebraic geometry]].<ref name="Shafarevich1994">The projective planes over fields are used throughout {{harvp|Shafarevich|1994}}, for example.</ref>

The [[quaternionic projective space|quaternionic projective plane]] '''HP'''<sup>2</sup> is also of independent interest.<ref>See, e.g., {{harvtxt|Weintraub|1978}} and {{harvtxt|Gorodkov|2019}}</ref>