1st Semester 2022/23: Topology in and via Logic

Instructors

Rodrigo N. Almeida, Nick Bezhanishvili, Soren B. Knudstorp

ECTS

6

Description

Topology is one of the basic areas of contemporary mathematics, and finds applications in all areas of logic: from traditionally mathematical subjects (model theory, set theory, category theory and algebraic logic) to areas of philosophy (epistemic logic, formal epistemology), formal semantics or computation (domain theory, learning theory). Its key idea is that one can understand space through very simple units - so called "open" and "closed" sets, and their interaction - in a way that can capture both the intuitive properties of physical space, and also more abstract notions of  "space": spaces of ideas, information spaces, or even spaces of actions, for example, in computation.

In this project we will familiarise students with the basic concepts of topology as they are used in logical practice. This will consist first in a series of introductory 1.5-hour lectures (including time for questions and discussion) covering the basic core concepts one tends to encounter in these settings (continuity, neighbourhood filters, compactness, connectedness, separation), as well as forays into more advanced topics and relationships, pursued by students. Importantly, we will emphasise how topology appears naturally in many logical contexts, and use them to develop intuition about the crucial concepts of topology.

Organisation

The lectures will be hybrid, although we do encourage in-person attendance. The (tentative) schedule is as follows:

Week 1:
3 lectures (Continuity, Neighbourhoods, Separation axioms)

Week 2:

3 lectures (Compactness, Connectedness, Basic topological constructions)

Week 3:

Individual consultation (to help develop presentations, and further research interests)

Week 4:

Presentations, to be scheduled at the end of week 2.

The topics of presentation will be suggested to students, and should focus on a more advanced concept or idea relating topology to logic. These can include (but are not limited to) specific aspects of the following broad topics: basic point-free topology; Polish spaces and the spaces of Cantor and Baire; Baire category theorems; Alexandroff and Boolean spaces; density and scatteredness. Students can also propose topics, so long as they broadly fit in the scope of the project.

Prerequisites

Basic mathematical maturity as can be expected from having taken a course heavily reliant on proofs, e.g. Philosophical Logic, Dynamic Epistemic Logic or Introduction to Modal Logic. Attendance of these courses can be useful to appreciate some of the examples, but is not necessary.
Basic familiarity with topics of mathematical logic will be useful in choosing presentation topics.

Assessment

Homework Assignments, Oral Presentation (week 4).

References

Ryszard Engelking, (1968) General Topology

Lecture notes will be made available throughout the course, with supporting material for the lectures.

More specific references will be made available during the project.