Topology
Topologi

FMAP15, 7.5 credits, A (Second Cycle)

Valid for: 2025/26
Faculty: Faculty of Engineering LTH
Decided by: PLED F/Pi
Date of Decision: 2025-04-10
Effective: 2025-05-05

General Information

Depth of study relative to the degree requirements: Second cycle, in-depth level of the course cannot be classified
Elective for: F4, F4-mtm, Pi4-mtm
Language of instruction: The course will be given in English

Aim

The main goal of the course is to introduce students to the principles and techniques of topology, and to enable them to apply these techniques in different mathematical settings. General topology is used in other contexts as well, e.g. within data analysis and physics, in situations in which one cannot (or do not want to) define shape or closeness between points in terms of a norm (or a metric).

Learning outcomes

Knowledge and understanding
For a passing grade the student must

• be able to give a detailed account of the fundamental topological concepts included in the course,
• be able to identify the most important theorems in the course and present their proofs,
• be able to explain the theory behind the methods introduced in the course,
• be able to give examples of a variety of topological spaces, and explain their relevance in different areas of mathematics,
• be able to recognise the topological structures in a range of mathematical problems.

Competences and skills
For a passing grade the student must

• be able to critically and systematically integrate knowledge from different areas of mathematics to analyse and solve complex problems using topological methods,
• be able to independently and creatively identify, formulate and solve relevant problems within the framework of the course.

Judgement and approach
For a passing grade the student must

• be able to argue for the important role of topological structures in different areas of research in mathematics and its applications,
• identify the own need for further knowledge and take responsibility for further knowledge development.

Contents

• Topological spaces and metric spaces with examples, product topologies, continuity of functions,
• Connectedness, completeness, and compactness, including the Arzela-Ascoli Theorem,
• Examples of applications and topological structures,
• Examples of metric and topological spaces relevant in other areas of mathematics, such as normed spaces and Hilbert spaces.

Examination details

Grading scale: TH - (U, 3, 4, 5) - (Fail, Three, Four, Five)
Assessment:

The examination consists of a written examination and a corresponding oral examination at the end of the course. The oral examination is only given to those students who have passed the written examination. Completed homework assignments can give a certain amount of bonus points that can be counted towards the written examination; this will be specified at the start of the course.

To pass on the course the student must pass -- i.e. have at least 50% of the maximal possible score -- on each of the exams. For the grade 4 at least 67% of the joint total are required, and for the grade 5 at least 80% of the joint total.

The maximum number of points that can be obtained on the written and the oral examinations are weighted three to one.

The examiner, in consultation with Disability Support Services, may deviate from the regular form of examination in order to provide a permanently disabled student with a form of examination equivalent to that of a student without a disability.

Modules
Code: 0125. Name: Written Examination.
Credits: 5.0. Grading scale: UG - (U, G). Assessment: Written test comprising theory and problems. The module includes: The whole course.
Code: 0225. Name: Oral Examination.
Credits: 2.5. Grading scale: UG - (U, G). Assessment: Oral examination of the theory.

Admission

Admission requirements:

• FMAF01 Mathematics - Analytic Functions and FMAF05 Mathematics - Systems and Transforms and FMAN55 Applied Mathematics

Reading list

• Munkres, James R: Topology. Upper Saddle River, N.J. Prentice Hall, 2000, ISBN: 0131816292. Main Textbook.
• Gamelin, Theodore W: Introduction to topology. Mineola, N.Y. : Dover Publications, 1999, ISBN: 0486406806. Alternative text book.
  https://ebookcentral.proquest.com/lib/lund/detail.action?docID=1893208
• Simmons, George Finlay, 1925-2019: Introduction to topology and modern analysis. New York : McGraw-Hill, 1963, ISBN: 0070573891. Complementary reading. Later editions exist.
• Steen, Lynn Arthur: Counterexamples in topology. New York : Dover Publications, 1995, ISBN: 048668735X. Complementary reading.

Contact

Teacher: Sandra Pott, sandra.pott@math.lu.se
Examinator: Anders Holst, Anders.Holst@math.lth.se
Director of studies: Anders Holst, Anders.Holst@math.lth.se
Director of studies: Anna-Maria Persson, anna-maria.persson@math.lu.se