## SE328 : Topology

Topology is a mathematical study on spaces and their invariant properties, and also relationships between other spaces. Not only it is important subject in pure Mathematics, but also it is applied to many areas such as applied mathematics (optimization), physics (quantum theory), chemistry (molecular structure), biology (DNA folding), ecology (behavioral ecology), computer science (visual analysis), mechanics (control theory), economics (fair division), and so on. In this course, we will learn

__Basic set theory__: the set of real numbers and integers, axioms of set theory, countable sets, well-ordered sets, maximum principle.__Topological spaces__: basis, product space, quotient space, metric space, limit points, closure, convergent sequence, continuous function__Connectedness__: local connectedness, path connectedness, connected subsets in the real line.__Compactness__: limit points in compact spaces, compact subspaces, compact subsets in the real line.__Countability and Urysohn lemma__: regular and normal lspaces, Urysohn lemma, Urysohn metrization theorem__Topological manifolds__: m-dimensional manifolds, partition of unity__Metric spaces__: complete spaces, space filling curves, topologies on function spaces__Homotopy of path__: homotopy, simply connected space, covering space__Fundamental groups__: basic group theory, the fundamental group of circle, the action of fundamental group on the covering spaces__Homotopy theory__: homotopy types, Brouwer fixed point theorem, fundamental theorem of algebra, Borsuk-Ulam theorem.