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.