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The classification theorem for compact surfaces is a formidable result. This result was obtained in the early 1920's, and was the culmination of the work of many. The theorem gives a simple way of obtaining all compact 2-manifolds, moreover, as a result of the theorem, it's possible to decide whether or not any two compact surfaces are homeomorphic rather easily. Before the statement of the theorem, quite a bit of basic topological concepts are first introduced, including connectivity, compactness and quotient topology. In addition to that, a rigorous proof requires, among other things, a precise definition of a surface, orientability, a notion of generalized triangulation, and a precise way of determining whether two surfaces are homeomorphic, which requires some notion of algebraic topology. All of the above brings together the final proof of the theorem.
Dianna Xu, "Classification Theorem for Compact Surfaces", . December 2001.
Date Posted: 20 June 2007