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Syllabus:

# Topology, 7.5 Credits

Swedish name: Topologi

This syllabus is valid: 2018-08-20 and until further notice

Course code: 5MA191

Credit points: 7.5

Education level: First cycle

Main Field of Study and progress level: Mathematics: First cycle, has at least 60 credits in first-cycle course/s as entry requirements

Responsible department: Department of Mathematics and Mathematical Statistics

Established by: Faculty Board of Science and Technology, 2018-06-21

## Contents

The course offers knowledge of notions and theorems in topology. The notions compactness and continuity are central. Functions defined between topological spaces are studied with emphasis on continuity. Central theorems are Urysohn's lemma, Tietze's expansion theorem and Tychonoff's theorem.

## Expected learning outcomes

For a passing grade, the student must be able to

Knowledge and understanding

• thoroughly account for definitions and theorems in set topology and metric spaces
• thoroughly account for the notions open set and continuous function

Skills and abilities

• independently construct topologic spaces and functions with given topological properties
• independently conduct formal proofs of theorems in measure topology
• independently decide the topological properties of different sets
• independently decide the properties of functions between topological spaces

## Required Knowledge

The course requires courses in Mathematics, minimum 60 ECTS or at least two years of university studies and both cases requires courses in Calculus, minimum 22,5 ECTS including a course in Multivariate Calculus minimum 7,5 ECTS, or equivalent.

## Form of instruction

Teaching is mainly in the form of lectures.

## Examination modes

The course is examined by a written exam. For the course, one of the following grades is assigned: Fail (U), Pass (G), Pass with distinction (VG).

A student who has received a passing grade on a test is not allowed to retake the test in order to receive a higher grade. A student who has not received a passing grade after participating in two tests has the right to be assigned another examiner, unless there are certain circumstances prohibiting this (see the Higher Education Ordinance, chapter 6, 22§). A request to be assigned another examiner should be addressed to the head of department for the department of mathematics and mathematical statistics. The possibility of being examined based on the current version of the syllabus is guaranteed for at least two years following the student's first participation in the course.

Credit transfer
All students have the right to have their previous education or equivalent, and  their working life experience evaluated for possible consideration in the corresponding education at Umeå university. Application forms should be adressed to Student services/Degree evaluation office. More information regarding credit transfer can be found on the student web pages of Umeå university, http://www.student.umu.se, and in the Higher Education Ordinance (chapter 6). If denied, the application can be appealed (as per the Higher Education Ordinance, chapter 12) to Överklagandenämnden för högskolan. This includes partially denied applications

## Other regulations

This course can not be included in a degree together with another course with similar contents. When in doubt, the student should consult the director of study at the department of mathematics and mathematical statistics.

## Literature

### Valid from: 2018 week 34

Munkres James R.
Topology
Second edition [2018 reissue]. : New York, NY : Pearson : [2018] : xvi, 537 pages :
ISBN: 9780134689517
Mandatory
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