Relation To The Real Projective Plane
The sphere, before being transformed, is not homeomorphic to the real projective plane, RP2. But the sphere centered at the origin has this property, that if point (x,y,z) belongs to the sphere, then so does the antipodal point (-x,-y,-z) and these two points are different: they lie on opposite sides of the center of the sphere.
The transformation T converts both of these antipodal points into the same point,
If this were true for only one or small subset of points of the sphere, then these points would just be double points. But since it is true of all points, then it is possible to consider the Roman surface to be homeomorphic to a "sphere modulo antipodes". However, these are not the only identifications that occur under this map. Consequently, the Roman surface is a quotient of the real projective plane RP2 = S2 / (x~-x). Furthermore, this quotient has the special property that it is locally injective, making this an immersion of RP2 into 3-space. It was previously stated that the Roman surface is homeomorphic to RP2, but this was in error.
Read more about this topic: Roman Surface
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