I am looking into formulating a good project description, for a special course at Advanced undergraduate or graduate level, in which I afterwards will at leat be able to do the following.
- Transform intuitive notions about continuity and convergence to exact mathematics.
- Construct mathematical proofs of a methodological character.
- Operate and argue with abstract notions of distances.
- Work with abstract notions of distance in the study of continuity and convergence.
- Operate and argue with abstract topological concepts.
- Work with abstract topological concepts in the study of continuity and convergence.
- Exploit knowledge about the topology of point sets in the study of extremal properties of continuous functions.
- Construct the completion of a metric space.
I think that topics within Real and Complex analysis, Operator Theory or Functional Analysis, is the most intriguing. Does anyone have an idea of which direction I should advance?