The real projective space represented by is the space obtained from under the equivalence relation . Basically, is the set of lines which passed through the origin in . It can also be understood by identifying antipodal points(points which lie on opposite ends of the diameter) of a unit -sphere,.
One very basic yet deeply interesting example of these spaces is , known as the real projective plane. While is a points and is homeomorphic to a circle with infinity, the real projective plane turns out to be far more interesting indeed. It can’t be embedded in and its immersion/s such as Boy’s surface, Roman surface and Cross Cap have far stranger structures than a mere circle as in the case of . In fact, I will delineate some of the calculations involved in the Kusner-Bryant 3-dimensional parametrization of Boy’s surface. It’s a little fascinating how much complexity can be added to mathematical structures upon generalization especially in the case of the projective space which I believe have a remarkably simple and ‘non-threatening’ definition.