MM509: Topology II (5 ECTS)

STADS: 13000901

Level
Bachelor course

Teaching period
The course is offered in the spring semester.
3rd quarter.

Teacher responsible
Email: swann@imada.sdu.dk

Timetable
Show entire timetable
Show personal time table for this course.

Comment:
Ubegrænset deltagerantal, dog højst 30 studerende pr. hold for eksaminatorierne.

Prerequisites:
None

Academic preconditions:
The content of the courses MM501 Calculus I, MM502 Calculus II and MM508 Topology I must be known.

Course introduction
Topological properties play an important role in most branches of mathematics. The purpose of the course is to develop further the concepts, which are introduced in Topology I, in more advanced settings.

Qualifications
The course presents a number of important concepts and techniques in the general theory for metric and topological spaces. In addition, the students will learn how knowledge of advanced topological notions can provide the solution to problems in other branches of mathematics; e.g. solutions of particular systems of differential equations and approximation of general continuous functions by specific classes of continuous functions (e.g. polynomials).

Having completed the course successfully the students can be expected to

• have a solid understanding of fundamental topological concepts such as: uniform continuity, compact sets in metric and topological spaces, the relationship between compactness and sequential compactness.

• be able to use their acquired knowledge in topology to solve advanced problems in other branches of mathematics.

Expected learning outcome
By the end of the course the student will be able to:

• answer written exercises in point-set topology and its applications within the scope of the course's syllabus
• formulate the written answers in a way that is mathematically correct and rigorous
• give an oral presentation of the statement and proofs related to any subject on a previously given list of topics within the course's syllabus
• formualte the oral presentation in a mathematically correct way
• answer supplementary questions from the teacher and external examiner on definitions and results from the course syllabus

Subject overview
1) Metric and topological spaces.
2) Compactness and sequential compactness.
3) Uniform continuity of functions.
4) The Riemann integral.
5) Completeness of metric spaces.
6) The Banach fixed-point theorem.
7) Spaces of continuous functions viewed as topological spaces.
8) Existence and uniqueness of solutions to certain types of differential equations.
9) Approximation of continuous functions, e.g. the Stone-Weierstrass theorem.
10) Normal topological spaces.

Literature

• Meddeles ved kursets start..

Syllabus
See syllabus.

Website
This course uses e-learn (blackboard).

Prerequisites for participating in the exam
None

Assessment and marking:
Oral exam. Written exam. External marking. Marks according to the Danish 7-scale.

The courses Topology I and Topology II are evaluated together for students who have taken both courses. The evaluation consists of a two-hour written exam and a 30 minutes oral exam (with 30 minutes preparation).

Expected working hours
The teaching method is based on three phase model.

Forelæsninger (32 timer). Eksaminatorier (18 timer).
Educational activities

Language
This course is taught in Danish.

Course enrollment
See deadline of enrolment.

Tuition fees for single courses
See fees for single courses.