In this paper we will discuss a number of important theorems in topology, in particular the Seifert-van Kampen theorem and the classification of compact surfaces. The goal of this paper is to give the reader some intuition behind the classification theorem by studying certain topological spaces such as the n-fold torus and the m-fold projective plane.
We begin by stating the Seifert-van Kampen theorem, which will serve as a foundation for the remaining results. This is followed by an in depth introduction to polygonal surfaces, labelling schemes and the resulting quotient spaces, in which we compute several fundamental groups and study some of their properties. These examples and results indicate that certain spaces can be viewed as building blocks for others. This relationship is explored further in the classification theorem of quotient spaces obtained via a proper labelling scheme of a polygonal region. To conclude, we observe that any compact space is homeomorphic to such a quotient space, finalizing the classification.
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