Oftentimes, topologies on finite sets are unjustly overlooked, most likely because a Hausdorff topology on a finite set is automatically discrete. However, if one also chooses to consider non-Hausdorff topologies, a whole new world opens up. As it turns out, finite topologies can be used to model weak homotopy type of any finite polyhedron. In this thesis we present a homotopy and a weak-homotopy classification of finite topological spaces.
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