Whitney Embedding Theorem:  This tells us we may embed any smooth n-dimensional manifold in R2n Compact Surface:  A compact surface is a 2-dimensional manifold, having no boundary. Every point of such a surface must have some neighborhood homeomorphic to a disc. Triangulation ⇒ Euler characteristic: X = v-e+f. It can be proven that any triangulation which is planar has Euler characteristic 2 (counting the infinite face) - used to show that the Euler characteristic is an intrinsic property of the surface. Knowing (a) - Euler characteristic and (b) - orientability, we can classify all compact surfaces.
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