TMA4190 Introduction to Topology - Spring 2026

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.

See the study handbook for more information.

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!

Exercises can be found in the lectures notes, at the end of each chapter.

Here are some additional exercises you can try.

Vote here for which Chapter 6 exercises you would like to see on the board.

• Lecture notes

Deviations from the course material: There are several small changes from the lectures notes. I take the convention that the intersection over an empty set is the power set. See the discussion on math.stackexchange for some more information.

Correspondingly, for us a sub-basis is one of the following equivalent definitions:

Definition: Let \( X \) be a topological space with topology \( \tau \). A subbase of \( \tau \) is defined as a subcollection \( \mathcal{S} \) of \( \tau \) satisfying one of the three following equivalent conditions:

1. \( \tau \) is the smallest topology containing \( \mathcal{S} \): any topology \( \tau' \) on \( X \) containing \( \mathcal{S} \) must also contain \( \tau \).
2. \( \tau \) is the intersection of all topologies on \( X \) containing \( \mathcal{S} \).
3. The collection of open sets consisting of all finite intersections of elements of \( \mathcal{S} \) forms a basis for \( \tau \).

We do not ask that the subbase covers \( X \). In practice, the 'natural' sub-bases we see do have this property however, so it is not something to worry about too much.

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.

Written exam. 4 hours. No printed or hand-written support material is allowed. A specific basic calculator is allowed.

• Emil Guntvedt
• Morten Haltbakk
• Kristian Paasche Roland