Course Description: | This course develops the basic ideas of topology using the example of metric spaces to illustrate and motivate the general theory. Topics include metric spaces, convergence, completeness, and the contraction mapping theorem; metric topology, and open and closed subsets; topological spaces, subspaces, and product spaces; continuous mappings and homeomorphisms; compact spaces; connected spaces; and Hausdorff spaces and normal spaces. Applications include the implicit function theorem, chaotic dynamical systems, and an introduction to Hilbert spaces and abstract Fourier series. |