Topology in and via Logic (2024)
Coordinated project, January 2024, ILLC.
Coordination: Rodrigo N. Almeida , Nick Bezhanishvili, Qian Chen.
Powered by: Nick Bezhanishvili.
Project 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 be done by allowing students to self-study six recordings of 1.5-hour lectures, covering the basic core concepts one tends to encounter in these settings (continuity, neighbourhood filters, compactness, connectedness, separation), and it will include 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.
References
Ryszard Engelking, (1968) General Topology.
Steven Vickers, (1996) Topology via Logic.
Jorge Picado, Aleš Pultr, (2012) Frames and Locales: Topology without Points.
Tai-Danae Bradley, Tyler Bryson, John Terilla, (2020) Topology: A Categorical Approach.
Lecture notes for the project will be updated throughout the project. You can find the most recent version here. We thank you for any typos you might notice!
Assignments
There will be three Homework Assignments, which will be posted in this page. Solutions can be submitted until one week after the end of the block (i.e., 9th of February). You can submit your solutions by e-mailing Qian Chen. You will be able to find the assignments here:
Registration and Presentations
Students are expected to form a team of between two and three students. Please indicate the team, and in due time, the topic, in this sheet: Google Doc.
You can find some suggested topics for presentations here.
Lecture Schedule
The lectures were recorded in the year 2024; their schedule then was as follows:
| Number | Topic | Location | Time |
|---|---|---|---|
| Zero | Kick-Off Meeting: Introductions. Basic Epistemic intuitions for Topology. | Online (slides) | N/A |
| First Lecture | Topological Spaces. The Epistemic Interpretation. Bases and Subbases. Examples of Topological Spaces. | L1.10 (slides) | 11h00-13h00 |
| Second Lecture | Examples of Topological Spaces (continued). Topological Constructions: Subspaces and Products. Neighbourhoods. Interior and Closure Operators. | B0.202 (slides) | 11h00-13h00 |
| Third Lecture | Continuous and Open maps. Continuity as Computability. Examples of Continuous functions. Restricting to Bases and Subbases. | G2.13 (slides) | 11h00-13h00 |
| Fourth Lecture | Introduction to Filters. Filter Convergence: Examples and Epistemic Motivation. Hausdorff Spaces. Equivalence of Hausdorff with Uniqueness of convergence of filters. Weaker Separation Axioms: T1 and T0. Stronger Separation Axioms: T4. | Seminar Room - F1.15 - (ILLC) (slides) | 11h00-13h00 |
| Fifth Lecture | Extending Filters: Prime filter theorem (without proof). Compactness. Finite Intersection Property. Equivalence of Compactness with existence of points for convergence. Compact Hausdorff spaces: Basic properties. Introduction to Compactifications. | Seminar Room - F1.15 - (ILLC) (slides) | 11h00-13h00 |
| Sixth Lecture | Alexandroff One-Point Compactification. Stone-Cech compactification. Connectedness of Topological spaces - epistemic and geometric motivations. Disconnectedness. Stone spaces. (slides) | B0.202 | 11h00-13h00 |
You will find the recordings of the lectures here.
Student Presentations
Below you will find the schedule for the presentations:
| Date | Time | Who | General Topic | Location | Notes |
|---|