Topology is the study of topological spaces, continuous maps between them, and properties preserved under continuous transformations. Topological spaces are ubiquitous in mathematics; Banach spaces, metric spaces, and varieties are all types of topological spaces.
In this course, we will start by introducing metric and general topological spaces, including basic properties like compactness and connectedness. We will then study elements of algebraic topology, including fundamental groups and covering spaces, homotopy and the degree of maps.
| Assignments | 30% |
| Midterm | 25% |
| Final exam | 45% |
Late problem sets will not be accepted except in the case of an emergency. At the end of the semester, your lowest problem set grade will be dropped from your average. This is meant to accommodate non-emergency absences, so try not to use this unless you have to.
| 09/02 | Introduction, metric spaces and topological spaces | Munkres, §12, §20 | Notes |
| 09/09 | Examples of topological spaces | Munkres, §13-16 | Notes |
| 09/16 | Continuous functions | Munkres, §17-18 | Notes |
| 09/23 | Connectedness | Munkres, §23-24 | Notes |
| 09/30 | Compactness | Munkres, §26-27 | Notes |
| 10/7 | Manifolds and complexes | Hatcher, §0 | Notes |
| 10/14 | The fundamental group | Hatcher, §1.1 | Notes |
| 10/21 | The fundamental group of the circle | Hatcher, §1.1 | Notes |
| 10/28 | Deformation retracts and homotopy equivalences | Hatcher, §0, §1.1 | Notes |
| 11/04--11/11 | The van Kampen Theorem | Hatcher, §1.2 | Notes (pages 1-8) |
| 11/18 | Covering spaces | Hatcher, §1.3 | Notes (pages 9-12) |
| 12/02-- | Classification of covering spaces | Hatcher, §1.3 | Notes |