Topology in and via Logic 2026

Description

Topology is one of the basic areas of contemporary mathematics, and plays a significant role in many areas of logic: from mathematical subjects (model theory, set theory, category theory, algebraic logic) to areas of philosophy (epistemic logic, formal epistemology), as well as formal semantics and theoretical computer science (domain theory, learning theory). The key idea behind topology is that spaces can be understood through simple building blocks – so-called “open” and “closed” sets – and their interactions. This framework captures not only the intuitive features of physical space but also a wide variety of abstract notions of “spaces,” such as spaces of ideals, information spaces, and spaces of actions.

This project introduces students to the concepts of topology as they are used in logical practice. It combines a series of introductory lecture recordings covering the basic concepts commonly used in the logical setting (continuity, neighbourhood filters, compactness, connectedness, separation) with Q&A sessions that focus on practical applications and help students select appropriate presentation topics. Students will also have the opportunity to explore more advanced topics according to their interests. Throughout the project, we will emphasize how topology appears naturally in many logical contexts.

Please email me (t.takahashi (at) uva.nl) for registration.

Organization

• Week 1: A brief kick-off meeting (online) + 3 lecture recordings + 1 tutorial
• January 5: kick-off meeting (13:00-13:30, online) (slides)
• January 9: tutorial 1 (11:00-13:00, F3.20)
• Homework 1 (due on January 15)
• Week 2: 3 lecture recordings + 1 tutorial
• January 16: tutorial 2 (11:00-13:00, F3.20)
• Homework 2 (due on January 22)
• Week 3: 2 Q&A sessions (preparation for the group presenation)
• January 19: Q&A session 1 (13:00-15:00, F3.20) (topic options)
• January 23: Q&A session 2 (11:00-12:00, F3.20)
• Week 4: Group presentation + Guest lecture by Nick Bezhanishvili
• January 27: Nick's lecture and group presentations (13:00-17:00, F1.15)
• January 29: group presentations (12:30-16:30, F1.15)

Tutorials, Q&A sessions, and group presentation will be in person.

Lecture recordings and notes

• Lecture notes (by Amity Aharoni, Rodrigo Nicolau Almeida, and Søren Brinck Knudstorp)
• Errata
• Lecture recordings from 2024

1. Topological Spaces. The Epistemic Interpretation. Bases and Subbases. Examples of Topological Spaces. (slides)
2. Examples of Topological Spaces (continued). Topological Constructions: Subspaces and Products. Neighbourhoods. Interior and Closure Operators. (slides)
3. Continuous and Open maps. Continuity as Computability. Examples of Continuous functions. Restricting to Bases and Subbases. (slides)
4. 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)
5. 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)
6. Alexandroff One-Point Compactification. Stone-Cech compactification. Connectedness of Topological spaces - epistemic and geometric motivations. Disconnectedness. Stone spaces. (slides)

Group presentation

• Some topic ideas

You are also welcome (and encouraged!) to explore your own topics based on your interests.

References

• Ryszard Engelking (1968). General Topology.
• Steven Vickers (1996). Topology via Logic.
• Jorge Picado, Ales Pultr (2012). Frames and Locales: Topology without Points.
• Tai-Danae Bradley, Tyler Bryson, and John Terilla (2020). Topology: A Categorical Approach.