# NMAA05010U Topology (Top)

Volume 2024/2025
Education

BSc Programme in Mathematics

Content

This is a course on topological spaces and continuous maps. Main topics of this course are:

• Topological Spaces
• Subspace, Order, Product, Metric and Quotient Topologies
• Continuous Functions
• Connectedness and Compactness
• Countability and Separation Axioms

Secondary topics are:

• Retractions and fixed points
• Tychonoff Theorem
• Compactifications
• Vistas of algebraic topology
Learning Outcome

This course will enable the participants to work with basic topological concepts and methods.  At the end of the course, the students are expected to have attained:

Knowledge:

• understand and assimluate the concepts and methods of the main course topics including basic definitions and theorems
• understand secondary topics covered in the specific course

Skills:

• determine properties of a topological space such as Hausdorffness, countability, (path) connectedness, (local) compactness
• construct new spaces as subspaces, quotient spaces and product spaces of known ones

Competences:

• analyze concrete topological spaces using acquired knowledge and skills
• relate the theory of topological spaces and continuous maps to specific settings in past and future math courses

Lebesgueintegralet og målteori (LIM) or similar
5 hours of lectures and 4 hours of exercises per week for 7 weeks.
• Category
• Hours
• Lectures
• 35
• Preparation
• 141
• Theory exercises
• 28
• Exam
• 2
• Total
• 206
Written
Oral
Individual
Continuous feedback during the course of the semester
Credit
7,5 ECTS
Type of assessment
Continuous assessment
Type of assessment details
Three sets of homework problems and a final 2 hour written in-class test in week 9 of the course. The final test must be passed with at least 30 points out of 100 as a prerequisite for passing the course. If this requirement is fulfilled, the homework and the final test each contribute 50% towards the final grade
Aid
Only certain aids allowed

Only certain aids allowed. All aids allowed for the homework; 2 pages of handwritten notes allowed for the final test.

Marking scale
Censorship form
No external censorship
One internal examiner
Re-exam

Consists of 2 elements:

1. A 2 hour written in-class test. Same requirements as for the ordinary exam. Only certain aids allowed: 2 pages of handwritten notes.
2. A 15 min oral examination (without preparation time) of the homework sets (weighted 50%). Students can choose to keep the results from their course work in which case they only participate in the written test, or they can participate in the 15 minutes oral examination, in which case the score from the oral examination will count as the homework.

If fewer than 10 people are signed up for the re-exam, the 2 hour written test will be replaced by a 20 min oral examination (without preparation). In this case, students who choose not to keep the results from their homework, will participate in a 35 minutes oral examination of which 20 minutes substitute for the written test, and 15 minutes are the examination of the 3 homework sets. The weights are 50% for homework and 50% for the oral test substituting the written test.

##### Criteria for exam assesment

The student should convincingly and accurately demonstrate the knowledge, skills and competences described under Intended learning outcome.