Duration
30h Th, 20h Pr, 10h Mon. WS
Number of credits
| Bachelor in mathematics | 6 crédits | |||
| Master in mathematics, research focus | 6 crédits | |||
| Master in mathematics, teaching focus (Réinscription uniquement, pas de nouvelle inscription) | 6 crédits |
Lecturer
Language(s) of instruction
French language
Organisation and examination
Teaching in the second semester
Schedule
Units courses prerequisite and corequisite
Prerequisite or corequisite units are presented within each program
Learning unit contents
This course is an introduction to general topology. Its main purpose is the abstract study of concepts such as continuity, compactness, and connectedness, together with the related properties and theorems. These notions, introduced in the first-year analysis course in the setting of Euclidean spaces, are generalized here to arbitrary sets endowed with a topology.
The topics covered include:
- Definition of a topology, neighbourhoods, interior, closure and boundary of a set, continuity of mappings;
- Order on the set of topologies, initial and final topologies, subspaces, product spaces and quotient spaces;
- Convergence in topological spaces, filters, separation and countability axioms;
- Compactness and connectedness, together with some classical theorems;
- Topological vector spaces.
Learning outcomes of the learning unit
By the end of the course, students will be able to present the theory covered in the course in a structured manner and apply it to solving exercises.
They will master the fundamental concepts of general topology.
They will also be able to study independently a topic related to the theory covered in the course and complete an assignment on a subject determined by the course instructor.
These fundamental notions of general topology will be useful throughout their studies, particularly in algebra, differential geometry, and analysis.
Prerequisite knowledge and skills
A basic knowledge of naive set theory, functions, Euclidean spaces, equivalence relations, and quotient sets is required. A good knowledge of basic topological notions in Euclidean spaces (open and closed sets, compactness, connectedness, etc.) is also required.
Planned learning activities and teaching methods
The course includes lectures, delivered either in person or online, as well as exercise sessions.
During the exercise sessions, the teaching assistant will solve some representative exercises on the board. Students will then regularly be asked to prepare individual exercises and present their solutions on the board.
Mode of delivery (face to face, distance learning, hybrid learning)
Blended learning
Further information:
The teaching arrangements will be specified depending on the number of students enrolled.
Course materials and recommended or required readings
Platform(s) used for course materials:
- eCampus
Further information:
The course notes are available on eCampus.
Many textbooks on general topology are also available in the Mathematics Library (Building B52).
Exam(s) in session
Any session
- In-person
written exam AND oral exam
Further information:
The exam consists of a written part and an oral part.
The written part focuses on solving exercises related to the topics covered in the lectures and exercise sessions.
The oral part covers the theory presented in the course and its immediate applications.
Work placement(s)
Organisational remarks and main changes to the course
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Contacts
Céline Esser (Celine.Esser@uliege.be)
Assistant : Pierre Stas