Coordinated project, January 2024, ILLC.

Coordination: Rodrigo N. Almeida , Søren B. Knudstorp, Amity Aharoni.

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).<br>

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.

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!

There will be three tutorials. Amity Aharoni will be in charge of designing the tutorial sheets and running the tutorials. You can find the tutorials here:

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 here. You can find the assignments here:

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.

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 |
---|---|---|---|

2024/05/01 | Kick-Off Meeting: Introductions. Basic Epistemic intuitions for Topology. | Online (slides) | N/A |

2024/08/01 | 1st Lecture: Topological Spaces. The Epistemic Interpretation. Bases and Subbases. Examples of Topological Spaces. | L1.10 (slides) | 11h00-13h00 |

2024/10/01 | Tutorial | L1.11 | 17h00-19h00 |

2024/12/01 | 2nd Lecture: Examples of Topological Spaces (continued). Topological Constructions: Subspaces and Products. Neighbourhoods. Interior and Closure Operators. | B0.202 (slides) | 11h00-13h00 |

2024/15/01 | 3rd Lecture: Continuous and Open maps. Continuity as Computability. Examples of Continuous functions. Restricting to Bases and Subbases. | G2.13 (slides) | 11h00-13h00 |

2024/16/01 | Tutorial | Room Next to the MoL Room | 17h00-19h00 |

2024/17/01 | 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. | Seminar Room - F1.15 - (ILLC) (slides) | 11h00-13h00 |

2024/18/01 | 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. | Seminar Room - F1.15 - (ILLC) (slides) | 11h00-13h00 |

2024/19/01 | 6th Lecture: Alexandroff One-Point Compactification. Stone-Cech compactification. Connectedness of Topological spaces - epistemic and geometric motivations. Disconnectedness. Stone spaces. (slides) | B0.202 | 11h00-13h00 |

2024/22/01 | Tutorial | Room Next to the MoL Room | 17h00-19h00 |

2024/24/01 | Tutorial | L1.11 | 17h00-19h00 |

2024/29/01 | Guest Lecture: Nick Bezhanishvili on Topological Semantics of Modal Logic | TBA | 12h00 |

2024/30/01 | Presentations 1 | Seminar Room - F1.15 - (ILLC) | 15h00-18h00 |

2024/31/01 | Presentations 2 | L1.11 | 15h00-18h00 |

You will find the recordings of the lectures here. These recordings will be deleted in April 2024.

You can find below the schedule for the presentations:

Date | Time | Who | General Topic | Location | Notes |
---|---|---|---|---|---|

30/01/2024 | 15h00-15h40 | Marco de Mayda, Jonathan Osser | Constructive Point-free Topology | Seminar Room (F1.15) | |

15h50-16h30 | Cezary Klamra, Kali Tolsma | Suslin’s Problem | |||

16h40-17h40 | Giacomo de Antonellis, Lucrezia Mosconi, Blaz Istenich Urh | Topological Games | |||

31/01/2024 | 15h00-15h40 | Kira Miller, Sid Singh | Epistemology and Topology | L1.11 | |

16h00-16h40 | Simone Killian, Tristan Hewitt | Point-free Topology and Duality | |||

17h00-17h40 | Lamarana Barrie, Gwan Yu Tijook | Topological Semantics of Logic |