Topology in and via Logic (2025)
Coordinated project, January 2025, ILLC.
Instructors: Qian Chen, Rodrigo N. Almeida and 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 familiarize students with the basic concepts of topology as they are used in logical practice. This will consist in a series of introductory lecture recordings covering the basic core concepts one tends to encounter in these settings (continuity, neighbourhood filters, compactness, connectedness, separation); Q&A sessions, guiding students through practical applications of the concepts, as well as guiding them in finding an appropriate presentation topic; as well as forays into more advanced topics, pursued by the students. Importantly, we will emphasize how topology appears naturally in many logical contexts.
To sign up for this project, send me an email :)
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.
More specific references will be made available during the project. You can find the current version of the lecture note here. We thank you for any typos you might notice!
Organization and Schedule
Students are encouraged to follow the recording, read the lecture notes and finish the corresponding assignments every week. There will be a tutorial and two Q&A sessions. The schedule is as follows:
- Week 1: A short online kick-off meeting + 3 lecture recordings + Tutorial
- Jan 7: Kick-off meeting. (Slides)
- Jan 10: Tutorial from 15:00 to 16:00, in Room L1.12, Lab 42. (Tutorial sheet)
- Week 2: 3 lecture recordings + Q&A
- Jan 13: The deadline of HW1, at 14:59.
- Jan 17: The deadline of HW2, at 14:59.
- Jan 17: Q&A session from 15:00 to 16:00, in Room L1.12, Lab 42.
- Week 3: Group consultations + Q&A
- Group consultations will be in this week.
- Jan 24: The deadline of HW3, at 14:59.
- Jan 24: Q&A session from 15:00 to 16:00, in Room L1.12, Lab 42.
- Week 4: Group presentations + guest lecture(s).
- Jan 27: Lecture on Modal Logic of Space by Nick at 14:00. Location: B0.204.
- Group presentations will be in this week.
Homework Assignments
The will be three homework assignments. To submit them, please send it to me at: q.chen2@uva.nl.
Tutorial
Here is the Tutorial sheet and answer to some of the exercises.
Group Presentations
Students are expected to form a team of 2-3 people and give a presentation on more advanced topics.
A list of possible topics can be found here.
The presentations will be on Wednesday and Friday in Week 4. The current schedule is as follows:
| Time | Presenters | General Topic |
|---|---|---|
| January 29 (Wed.) | ||
| 14:30 - 15:00 | Milan Hartwig, Victor Joss | Deforming Space |
| 15:10 - 15:40 | Vighnesh Iyer, Pim Amorison, Estel Koole | Weighing the character of a space |
| 15:50 - 16:20 | Louise Wilk, Andy Arteaga | Topology of Surprise |
| January 31 (Fri.) | ||
| 14:00 - 14:30 | Yiwei Liang, Shengmen Luo, Guannan Mi | General Topology: Metric Spaces |
| 14:40 - 15:10 | Ageno Ludovico, Menorello Edoardo | It is simple[x] |
| 15:20 - 15:50 | Marco de Mayda, Timo Franssen, Matteo Celli | Space is a Soup |
| 16:00 - 16:30 | Merlin Krzemien, Dor Lotan | Misleading Defeaters and Defeating Misleaders |
Lecture Recordings from Last Year
You can find the recordings of the lectures here. There are 6 lectures. The contents and corresponding slides are as follows:
- Topological Spaces. The Epistemic Interpretation. Bases and Subbases. Examples of Topological Spaces. (Slides)
- Examples of Topological Spaces (continued). Topological Constructions: Subspaces and Products. Neighbourhoods. Interior and Closure Operators. (Slides)
- Continuous and Open maps. Continuity as Computability. Examples of Continuous functions. Restricting to Bases and Subbases. (Slides)
- 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. (Slides)
- 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. (Slides)
- Alexandroff One-Point Compactification. Stone-Cech compactification. Connectedness of Topological spaces - epistemic and geometric motivations. Disconnectedness. Stone spaces. (Slides)
