The general topology allows to adapt the notions of continuity and convergence to the needs of a mathematical problem. This course is an introduction to the basic notions of topology, which are essential to follow a functional analysis course.
Fundamental properties (open and closed, neighbourhoods, convergence of a sequence), Bases, Subbases and Local bases (fundamental neighbourhoods). Continuous Functions (global, local and sequential continuity at a point) and Equivalent Topological Spaces, Denumerability (use of sequences) and Separation (uniqueness of a limit). Compactness (open covering and extraction of a finite subcovering) (Alexander and Heine-Borel theorems, properties) and Sequential Compactness (extraction of a convergent sub-sequence). Product spaces (product topology, Tychonoff compactness theorem). Metric spaces (metric topology, compactness in metric spaces (equivalence with sequential compactness), complete metric spaces).
The examination consists of two parts: a written examination and an oral examination. The written examination questions are exercise questions only: they deal with applications of the same kind as those proposed in the tutorials and the course. They aim to assess the student's ability to apply the main concepts and results of the course. For the oral examination, the questions focus on theory. The emphasis is on understanding, accuracy and synthesis. The exam consists of two questions, one typically consisting of stating a result and demonstrating it in context, the other being a synthesis question which consists of presenting a topic as concisely and completely as possible, without demonstration. The student has one hour of preparation time for these two questions. The presentation of a question, including the answers to the sub-questions, may never exceed fifteen minutes. This unit consists of three learning activities, each with a corresponding grade. The written and oral examinations each count for 40% of the final grade. The course is completed by a personal project which counts for 20% of the final grade. Precise instructions will be given in due course.
- Adams, Colin and Robert Franzosa, Introduction to topology: pure and applied, Pearson Prentice Hall, Upper Saddle River, NJ, 2008. [BUMP: 2710] - Lipschutz, Seymour, Topology - Courses and Problems, Schaum Series, McGraw-Hill, New York 1965, Paris (for the French translation) 1981. [BUMP: SB08413/006]
| Training | Study programme | Block | Credits | Mandatory |
|---|---|---|---|---|
| Bachelor in Computer Science | Standard | 0 | 4 | |
| Bachelor in Computer Science | Standard | 2 | 4 |