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CLASSIFICATION OF SURFACES

The sphere, torus, Klein bottle, and the projective plane are the classical examples of orientable and non-orientable surfaces. As with much of mathematics, it is natural to ask the question: are these all possible surfaces, or, more generally, can we classify all possible surfaces? In this paper, we examine a result originally due to Seifert and Threlfall that all compact surfaces are homeomorphic to the sphere, the connect sum of tori, or the connect sum of projective planes; for this paper, we follow a modern proof from Lee [2].