Coordinated project, January 2023, ILLC.
Coordination: Rodrigo N. Almeida , Soren B. Knudstorp.
Powered by: Nick Bezhanishvili.
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.
Ryszard Engelking, (1968) General Topology.
Steven Vickers, (1996) Topology via Logic.
Jorge Picado, Aleš Pultr, (2012) Frames and Locales: Topology without Points.
Lecture notes for the project will be made available and updated throughout the project. You can find the most recent version here.
Special thanks to Amity Aharoni for contributing to these lecture notes.
There will be two Homework Assignments, which will be posted in this page. Solutions can be submitted until one week after the end of the block (i.e., 10th of February). You can submit your solutions here. You can find the assignments here:
There will be three small quizzes. Once available, you can find the link to the forms below. They will be made available after Lectures 1, 3 and 5:
If you have not already, please indicate in this Google Doc whether you are taking the project for credit or not. In case you are taking it for credit, please form a team of between two and three students, and indicate there the team.<br>
You can find some suggested topics for presentations here.
There will be six lectures and a guest lecture throughout the project. These will take place in the following schedule, and will be broadcast online via this link:
| Date | Topic | Location | Time |
|---|---|---|---|
| 2023/04/01 | Kick-Off Meeting: Introductions. Basic Epistemic intuitions for Topology. | Online (slides) | N/A |
| 2023/10/01 | 1st Lecture: Topological Spaces. The Epistemic Interpretation. Bases and Subbases. Examples of Topological Spaces. | D1.111 (slides) | 15h00-17h00 |
| 2023/11/01 | 2nd Lecture: Examples of Topological Spaces (continued). Topological Constructions: Subspaces and Products. Neighbourhoods. Interior and Closure Operators. | D1.115 (slides) | 15h00-17h00 |
| 13/1/2023 | 3rd Lecture: Continuous and Open maps. Continuity as Computability. Examples of Continuous functions. Restricting to Bases and Subbases | D1.115 (slides) | 13h00-15h00 |
| 17/1/2023 | 4th 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. | G2.10 (slides) | 15h00-17h00 |
| 18/1/2023 | 5th 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. | D1.115 (slides) | 15h00-17h00 |
| 20/1/2023 | 6th Lecture: Alexandroff One-Point Compactification. Stone-Cech compactification. Connectedness of Topological spaces - epistemic and geometric motivations. Disconnectedness. Stone spaces. | D1.115 (slides) | 13h00-15h00 |
| 24/1/2023 | Guest Lecture | Online (slides) | 13h00-15h00 |
| 31/1/2023 | Presentations 1 | G0.05 | 15h00-18h00 |
| 2023/01/02 | Presentations 2 | D1.114 | 14h00-17h00 |
| 2023/03/02 | Presentations 3 | D1.113 | 13h00-16h00 |
You can find the recordings here (unfortunately the first two recordings do not show the board; this was fixed in future recordings). These recordings will be deleted in April 2023.
Information for the presenters: please send your slides to Rodrigo or Soren at least 30 minutes before the presentation.
| Date | Time | Who | General Topic | Location | Notes |
|---|---|---|---|---|---|
| 31/01 | 15h00-15h50 | David Alvarez Lombardi Brendan Dufty Alexander Lind | Stone Duality (slides) | G0.05 | |
| 16h00-16h35 | Hannah van Santvliet Minke Vervweij | Topological Graph Theory (slides) | |||
| 16h45-17h35 | Elias Bronner Jonathal Thul Wouter Vromen | Epistemology and Topology (slides) | |||
| 01/02 | 14h00-15h05 | Sarah Dukic Alyssa Reynaldi Joel Saarinen Paulius Skaisgiris | Domain Theory (slides) | D1.114 | |
| 15h15-16h05 | Justus Becker Gabrielle Brancati Carella Xiaoshuang Yang | Topological Semantics of Modal Logic (slides) | |||
| 16h15-16h35 | Ramon de Villegas | Mereotopology (slides) | |||
| 03/02 | 13h00-14h05 | Josef von Hoffman Michael Muller Stefan Zotescu Steffano Zuffi | Conceptual Spaces | D1.113 | |
| 14h15-14h50 | Ferreol Lavaud Paul Talma | Suslin’s Problem | |||
| 15h00-15h35 | Laura Hernandez Martin Antoine Mercier | Mereology of the Intuitionistic Continuum (slides) |