German original

# Topology (dt. Topologie)

 Level, degree of commitment in original study programme Intermediate module, compulsory elective module Forms of teaching and learning,workload Lecture (4 SWS), recitation class (2 SWS), 270 hours (90 h attendance, 180 h private study) Credit points,formal requirements 9 CP Course requirement: Successful completion of at least 50 percent of the points from the weekly exercises. Examination type: Written or oral examination Language,Grading German,The grading is done with 0 to 15 points according to the examination regulations for study course B.Sc. Mathematics. Original study programme B.Sc. Mathematik / Mathematik Wahlpflichtmodule Duration,frequency One semester, Regelmäßig im Wechsel mit anderen intermediate moduleen der Geometrie Person in charge of the module's outline Prof. Dr. Ilka Agricola, Prof. Dr. Pablo Ramacher, Prof. Dr. Volkmar Welker

## Contents

• Fundamentals of set-theoretic topology: Open sets, continuous mappings.
• Bases, construction of topological spaces, connectivity, separation properties
• Compactness and metrizability: Central theorems on compactness,
• metrizability conditions
• Homotopy,: homotopy classes and equivalence, mappings of and in spheres
• Coverings: lifting properties, fundamental group

## Qualification Goals

The students

• understand basic principles of topological structures and recognize that such structures can be found in many parts of mathematics,
• practice the axiomatic approach and train their abstraction skills,
• develop a deeper understanding of the implications of elementary conditions on a topological space,
• practice mathematical methods (development of mathematical intuition and its formal justification in proofs),
• improve their oral communication skills in the recitation class by practicing free speech in front of an audience and during discussions.

## Prerequisites

None. The competences taught in the following modules are recommended: either Foundations of Mathematics and Linear Algebra I and Linear Algebra II or Basic Linear Algebra, either Analysis I and Analysis II or Basic Real Analysis.

• tom Dieck, Tammo: Topologie. Walter de Gruyter, 2000.
• Jänich, K.: Topologie, Springer 2001.
• Schubert, H.: Topologie, Teubner 1975.