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Real projective plane
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=== The projective sphere === Consider a [[sphere]], and let the [[great circle]]s of the sphere be "lines", and let pairs of [[antipodal point]]s be "points". It is easy to check that this system obeys the axioms required of a [[projective plane]]: * any pair of distinct great circles meet at a pair of antipodal points; and * any two distinct pairs of antipodal points lie on a single great circle. If we identify each point on the sphere with its antipodal point, then we get a representation of the real projective plane in which the "points" of the projective plane really are points. This means that the projective plane is the quotient space of the sphere obtained by partitioning the sphere into equivalence classes under the equivalence relation ~, where {{nowrap|''x'' ~ ''y''}} if {{nowrap|1=''y'' = ''x''}} or {{nowrap|1=''y'' = −''x''}}. This quotient space of the sphere is [[homeomorphic]] with the collection of all lines passing through the origin in '''R'''<sup>3</sup>. The quotient map from the sphere onto the real projective plane is in fact a two sheeted (i.e. two-to-one) [[covering map]]. It follows that the [[fundamental group]] of the real projective plane is the cyclic group of order 2; i.e., integers modulo 2. One can take the loop ''AB'' from the figure above to be the generator.
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