General topology

Duration

30h Th, 20h Pr, 10h Mon. WS

Number of credits

Lecturer

Céline Esser

Language(s) of instruction

French language

Organisation and examination

Teaching in the second semester

Schedule

Schedule online

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.
The main purpose of general topology is the abstract definition and study of concepts such as continuity of mappings, connectedness, compactness ...
These concepts are usually defined in the first course in analysis for Euclidean spaces. They will be generalized for arbitrary sets.
For instance, the following topics could be presented :
The general definition of a topology, neighborhoods of points, interior, closure and boundary of a set.
We will study the continuity of mappings and define the initial and final topologies.
We will deal with subspaces, product spaces and quotient spaces.
The axioms of separation will be studied.
The last sections will be devoted to the definitions and properties of compact spaces and connected spaces.
A few classical theorems will also be presented.

Learning outcomes of the learning unit

At the end of the course, the students should be able to make a presentation of the theory or to use it in order to solve exercises
They should also be able to read the literature in order to make a report and a short talk about a topic proposed by the teacher.
This knowledge of general topology is useful for the students for lectures in advanced algebra, geometry and analysis. It should also help them to think of several difficult points they could have to teach in the secondary school.

Prerequisite knowledge and skills

A basic knowledge of naïve set theory, functions, Euclidean spaces and quotient spaces is assumed. A good knowledge of topological concepts (open sets connectedness compactness) in the Euclidean spaces is useful.

Planned learning activities and teaching methods

The theory is given mainly on the blackboard and is illustrated by simple examples whenever it is possible. Students are encouraged to ask questions and to participate.
During the exercise sessions, students are asked to solve exercises under the supervision of the teacher, who also gives a solution. When possible, a list of exercises will be given a few days before the exercise session, so that the student can prepare the exercises before the session.

Mode of delivery (face to face, distance learning, hybrid learning)

Face-to-face
The schedule of the lectures is determined by the "conseil des études en mathématiques" and available on the website of the department of mathematics.

If the number of students is lower to 5, the course may be delivered as a distance-learning course and the students may be asked to provide a personal work. 

Organisational adjustments related to the current health context

Recommended or required readings

Lecture notes are available on the web page :
http://www.geodiff.ulg.ac.be

There are also many textbooks on general topology available in the library of mathematics (building B52).

Assessment methods and criteria

Below you will find information on the evaluation methods planned for in-person and remote exams as well as those planned for hybrid sessions. Depending on how the health crisis evolves, the chosen method will be communicated to you no later than one month before the start of the exam session.

Any session :

- In-person

written exam ( open-ended questions ) AND oral exam

- Remote

written exam ( open-ended questions ) AND oral exam

- If evaluation in "hybrid"

preferred in-person


Additional information:

A written examination of exercises will be organized.
There will also be an oral examination and the presentation of a personal work. 

Work placement(s)

Organizational remarks

The exam should take place in January, as indicated above. However, this has to be confirmed by the "Conseil des études en Mathématiques". This council also determines the schedule of exams.

Contacts

Céline Esser

Email : Celine.Esser@uliege.be 
Département de Mathématique, Allée de la Découverte, 12, B37, 4000 Liège, Sart-Tilman. Office 0/62