Academic Year 2021-2022



Dikran Dikranjan
Anna Giordano Bruno
Unit Credits
Teaching Period
Second Period
Course Type
Prerequisites. First two years of first-level degree in Mathematics
Teaching Methods. Lectures, exercises, seminars.
Verification of Learning. In order to assess the knowledge and skills acquired by the students during the course, with respect to the learning aims, the exam consists in an oral test. The students attending the course have the possibility to partially or completely replace the oral test with seminar activity based on topics related to the course.
More Information. Moodle platform (e-learning).

If necessary, the lessons can be given in English.


Knowledge and understanding

Know concepts and fundamental results of general topology and topological algebra, and some modern problems by noting the intrinsic difficulties.

Use a modern formal language in the formulation of topological problems.

Applied knowledge and understanding

Face and solve with an appropriate language some classical problems in topology.

Identify relations between topological questions and problems or theory from other areas of mathematics.

Solve topological problems, also beyond those treated in the course.

Cross-sectoral skills/soft skills

Autonomy of judgment

Identify techniques from algebra, set-theory, analysis or geometry, suitable to solve the assigned problems.

Evaluate the difficulties of specific problems in topology.

Communication skills

Communicate the arguments in general topology or topological algebra learned in the course.

Communicate to a non-specialized audience the main aspects to the classical theory of topological groups and some modern problem.

Learning skills

Be able to read a research article in topology concerning the arguments of the course.

Autonomously work in the bibliographical search.

Face the proposed problem, selecting the more meaningful ones.

The course provides a classical introduction to the fundamental aspects of general opology and topological algebra. The first part concerns the basic concepts and results of the theory of metric spaces and topological spaces, with special attention paid to compactness, connectedness, metrization and dimension. The main topic in the second part are the topological groups, in particular Pontryagin duality and applications to functional analysis, number theory and dynamical systems. A primary aim of the course is the development of techniques from algebra, set theory, geometry and analysis, learned in the fundamental courses, to be applied in the study of topological problems.
L. Außenhofer, D. Dikranjan, A. Giordano Bruno:

Introduction to Topological Groups and the Pontryagin-van Kampen Duality.