Projects in Previous Years

1st Semester 2007/08: Pointless Topology

Raul Leal Rodriguez
If you are interested in this project, please contact Raul by e-mail.

It is a common belief that there is always a point to study topology. Geometrical intuitions were one of the moving engines in the development of topology. Abstraction of the properties of Euclidean ('real') spaces dominated work on topology a long time. This approach leads to consideration of Hausdorff spaces as the most basic objects to study.

In recent developments, since around the 1980's, it has been seen that there are a lot interesting spaces that are not Hausdorff. Those spaces appear in other branches of mathematics different from analysis, for example in logic. These spaces motivated a more abstract study of topology where the emphasis was not on the space but on its open sets. This so-called "pointless" topology is a theory of information. An advantage is that no previous knowledge of real analysis is required.

The goal of this project is to give a basic introduction to topology as information. This introduction will be done using as main sources the book Topology via Logic by Steven Vickers and some papers about Formal Topology; we intend to cover the first eight Chapters of Vickers book. In the first weeks there will be lectures on the basic material. In the last weeks students will concentrate on investigating one particular topic. At the end of the project the students should have a basic overview of topology as a theory of information, and should have a deeper knowledge in some specific aspect of topology.

The evaluation will be roughly 40% assignments, 60% final paper. Some suggested subjects for the final paper:
  • Some chapter of Vickers's book not covered in the lectures.
  • A constructive proof of the Heine-Borel Theorem.
  • Stone duality.
  • Tychonoff without choice?
  • Formal Topology and Modal Logic (Chu Spaces)
  • Grothendieck topology.