TMA4190 Introduction to Topology - Spring 2020

Dear all,

Updated May 25, 2020, 1:30pm

You should all have received an email from NTNU explaining the change of format and grades for the exam. Find the same information under the Exam section.

Please note the following:

  • The exam will be a digital home exam given on May 27, 9.00 - 13.00 (9am - 1pm).
  • Grades have been changed from A - F to pass/fail.
  • The exam will consist of a mixture of multiple choice problems and problems requiring you to show your work (see also the Exam section). Note that you will not be penalized for incorrect answers for the multiple choice problems.
  • The syllabus will consist of chapters 1 through 8 in the lecture notes. The intended material for chapter 9 will not be part of the syllabus.
  • Trial exam with solutions. Try working through the entire exam before looking at the solutions. (There was a misprint in a previous version of the printout from Inspera Assessment for problem 4 (it should read as in the solutions with \(A = \{x\in \mathbb{Q} \mid -\sqrt 5 < x < \sqrt 5\}\) and not \(A = \{x\in \mathbb{Q} \mid -\sqrt 5 < x < 5\}\)). Also note that you must upload an individual file for each non-multiple choice problem.
  • I have also written down some solutions to the remaining problems (from chapter 7 and 8) in the lecture notes.
  • I have not been able to find a satisfactory technical solution for a "digital office hour" before the exam. You are of course most welcome to send me an email about questions you might have regarding the course. Please refrain from sending me pictures with your handwriting so that your anonymity is preserved when doing the grading.

Best wishes,

Schedule Room
Lectures: Wednesday 12.15 - 14.00 EL4
Friday 08.15 - 10.00 EL4
Exam: Written exams To be announced
Marius Thaule
Office: 1248 Sentralbygg 2
Email: marius [dot] thaule [at] ntnu [dot] no

What this course is about

This course is a first introduction to topology. It will start with basic concepts in point set topology (e.g. topological spaces, continuous maps, metric spaces, constructions of topological spaces, compactness, connectedness). Then it will give an introduction to algebraic topology, such as homotopy, fundamental group, and covering spaces. A short introduction to homology will give an outlook on computational methods.

See the study handbook for more information.

What you need to know before this course

You should have seen multivariate calculus and linear algebra. Ideally you should have taken TMA4150 Algebra and/or MA3201 Rings and Modules as well, but if you haven't don't worry!

If you have any questions, just contact me!

Lecture Plan

The first lectures will be January 8 and 10.

Lecture Date Topic Notes References
1.1 08.01 Introduction, followed by an introduction to metric spaces Chapter 1 and 2
1.2 10.01 From continuous maps between metric spaces to topological spaces Chapter 2 and 3
2.1 15.01 Examples of topological spaces and continuous maps between topological spaces Chapter 3
2.2 17.01 Continuous maps between topological spaces, homeomorphisms and closed sets Chapter 3
3.1 22.01 Generating topologies Chapter 4
3.2 24.01 Generating topologies. Exercises (from chapter 2) Chapter 2 and 4
4.1 29.01 Exercises (from chapter 2 and 3). Constructing spaces Chapter 2, 3 and 5
4.2 31.01 Constructing spaces Chapter 5
5.1 05.02 Constructing spaces Chapter 5
5.2 07.02 Constructing spaces Chapter 5
6.1 12.02 Constructing spaces. Exercises (from chapter 4) Chapter 4 and 5
6.2 14.02 Exercises (from chapter 4). Topological properties Chapter 4 and 6
7.1 19.02 Postponed
7.2 21.02 Topological properties Chapter 6
8.1 26.02 Topological properties Chapter 6
8.2 28.02 Topological properties Chapter 6
9.1 04.03 Topological properties Chapter 6
9.2 06.03 Exercises (from chapter 5). The fundamental group Chapter 5 and 7
10.1 11.03 The fundamental group Chapter 7
10.2 13.03 The fundamental group Chapter 7
11.1 18.03 The fundamental group Chapter 7
11.2 20.03 The fundamental group Chapter 7
12.1 25.03 Exercises (from chapter 6). The fundamental group of the circle Chapter 6 and 8
12.2 27.03 The fundamental group of the circle Chapter 8
13.1 01.04 The fundamental group of the circle Chapter 8
13.2 03.04 The fundamental group of the circle Chapter 8
14.1 15.04 Exercises (from chapter 7 and 8). The Seifert-van Kampen theorem*
14.2 17.04 The Seifert-van Kampen theorem
15.1 22.04 The Seifert-van Kampen theorem. Exercises

* Not examinable.

Course material


We will not follow any particular textbook.

Some interesting books:

  • [A] M.A. Armstrong, Basic Topology, Springer-Verlag, 1983.
  • [Croo] F.H. Croom, Basic Concepts of Algebraic Topology, Springer-Verlag, 1978.
  • [Cros] M. Crossley, Essential Topology, Spring-Verlag, 2005.
  • [H] A. Hatcher, Algebraic Topology, Cambridge University Press, 2000.
  • [Ma] J.P. May, A Concise Course in Algebraic Topology, Chicago Lectures in Mathematics, 1999.

Some books on general topology:

  • [J] K. Jänich, Topology, Springer, 1984.
  • [Mu] J.R. Munkres, Topology: a first course, Prentice-Hall, 1975.


Informasjon om eksamen

Informasjon om eksamen

NTNU har vedtatt at det ikke skal arrangeres eksamen med fysisk oppmøte i april/mai/juni 2020. En rekke emner har på grunn av dette fått endret vurderingsform. En del emner har i tillegg endret karakterregel fra bokstavkarakter til bestått/ikke bestått.

TMA4190 - Introduksjon til topologi har endret vurderingsform. Vurderingen Skriftlig eksamen er endret til Hjemmeeksamen.

Karakterregel i emnet er endret til bestått/ikke bestått.

Emnebeskrivelsen og informasjon om vurdering i studentweb vil bli oppdatert innen 8.4.2020

Du vil finne mer informasjon om

  • Verktøy for eksamensgjennomføring (Inspera)
  • Forbedring av karakter ved endring av karakterregel
  • Trekkfrist
  • Konsekvens av endring av karakterregel fra bokstavkarakter til bestått/ikke bestått
  • Eksamensdato og varighet
  • Tilrettelegging
  • Hjelpemidler hjemmeeksamen
  • Utsatt eksamen
  • Egenmelding ved fravær på eksamen
  • Klage/begrunnelse på karakter
  • Ofte stilte spørsmål om eksamen og Korona
  • Faglig informasjon om eksamen

på denne nettsiden:

Sitter du fortsatt med ting du lurer på? Ta kontakt med instituttet på NTNU Hjelp.

Lykke til med vårens eksamensperiode!
Med vennlig hilsen,
Mersiha Sehic
Kontorsjef – Institutt for matematiske fag

Information about the exam

Information about the exam

NTNU has decided that no exams with physical attendance will be held in April/May/June of 2020. Because if this, several subjects have changed their form of assessment. Some of these have also changed their form of grading from letter grades to pass/fail.

TMA4190 - Introduction to Topology has a changed form of assessment. The assessment form Written examination has been changed to Home exam.

The form of grading is changed to pass/fail.

The course description and information regarding assessment will be updated in StudentWeb by April 8th 2020.

You will find more information concerning:

  • Tool for exam implementation (Inspera)
  • Improvement of grade and change in form of grading
  • Deadline for cancellation of exam registration
  • Consequences of change in form of assessment from letter grading to pass/fail
  • Exam dates and duration
  • Special needs accomodation
  • Examination aids for home examination
  • Re-sit exam
  • Self-certification for absence from assessment
  • Explanation of grades and appeals
  • FAQ - Frequently asked questions regarding exams and Corona
  • Academic information about exams

on this website:

If you still have questions, please contact the department using NTNU Help.

Good luck with the spring exam period!
With best regards,
Mersiha Sehic
Head of Office - Department of Mathematical Sciences

The exam will consist of a mixture of multiple choice problems and problems that will require you to show your work. For problems that require you to show your work you can, e.g., write on a regular piece of paper and then either scan it or photograph it and upload it to your exam in Inspera Assessment. For further information about Inspera Assessment, please consult innsida.

See also here for information regarding digital home exams.

The syllabus for the exam will consist of chapters 1 through 8 in the lecture notes (including exercises).

I have prepared a trial exam for you to practice on. Try to work through it from start to finish without looking at the solutions.

Reference Group

2020-05-25, Marius Thaule