ABSTRACT
The projective plane plays a fundamental role in geometry, and also in topology and algebraic geometry. In physics, the need for eliminating a Euclidean ambient space boasts a more colorful history. Though the normal Euclidean geometry remained valid on the small scale, namely, doing geometry on a flat surface, methods no longer sufficed when considering the geometry of the earth as a whole. As one moves around on the manifold, one passes from one coordinate patch to another. From a purely set-theoretic perspective, it is easy to define functions between manifolds. Despite the flexibility of the definition of a differentiable manifold, it does not allow for a boundary. Direct generalization to a similar theorem for regular surfaces, the Regular Value Theorem provides a class of examples of manifolds for which it would otherwise take a considerable amount of work to verify that the sets are indeed manifolds.
