# TMA4190 Introduction to Topology - Spring 2024

Schedule Room
Lectures: Monday 12.15 - 14.00 GL-RFB R3
Tuesday 8.15 - 10.00 GL-ELB EL4
 Instructor: Fernando Abellán https://www.ntnu.no/ansatte/fernando.a.garcia 1204 Sentralbygg 2 fernando [dot] a [dot] garcia [at] ntnu [dot] no

Topology its a vast field of study in mathematics: Informally speaking, topology studies the general notion of "space" or "shape" and gives a foundational framework for doing geometric constructions. Topology can be seen everywhere: From algebra to analysis and even in applied mathematics and physics.

This course is intented a first introduction to topology. We will define the fundamental notions of study such as topological spaces and continuous maps among them. We will spend some time learning which properties a topological space can satisfy (compactness, connectedness,countability and many more!) and what are the main constructions that can be perfomed with topological spaces (products, quotients,etc).

By the end of the course we will make a short introduction to algebraic topology: We will define the fundamental group of a topological space and perfom elementary computations.

## What you need to know before this course

You should have seen multivariate calculus and linear algebra. Some abstract algebra knowledge would be ideal (TMA4150 Algebra and/or MA3201 Rings and Modules) for the last part of this course. However, this is not mandatory and I will take care to define the necessary algebraic notions when the time comes.

If you have any questions, feel free to send me an email!

## Lecture Plan

The first lectures will be January 8 and 9.

 Lecture Date 0.1 08.01 Introduction. Metric spaces and continuous functions. 0.2 09.01 Metric spaces: Continuity in terms of open sets. 1.1 15.01 Topological spaces: Definition, continuous maps, open and closed sets 1.2 16.01 Topological spaces: Examples of continuous maps, closure/interior/boundary of a subset. Homeomorphisms ex.0 22.01 Solutions to exercise sheet 0 1.3 23.01 Homeomorphism as equivalence relation, Generating Topologies: Basis, subbasis and topology generated by a collection of subsets 2.1 29.01 Subspace Topology and its universal property. 2.2 30.01 Product Topology and its universal property 2.3 4.02 Solutions to exercise sheet 1 2.4 5.02 Quotient Topology, closed and open maps 2.5 12.02 Solutions to exercise sheet 2 3.1 13.02 Properties of Topological spaces: Hausdorff spaces, Compact spaces (to be continued) 3.2 19.02 Properties of Topological spaces: Compact spaces: Product of compact spaces is compact 3.3 20.02 Closed intervals are compact, Heine-Borel, connected spaces first definitions 3.4 26.02 Solutions to exercise sheet 3 3.5 27.02 Connected spaces, product of connected spaces is connected 3.6 4.03 Solutions to exercise sheet 4 4.1 5.03 Homotopies between continuous maps. 4.2 11.03 Concatenation of paths, associativity, unitality and inverses up to homotopy 4.3 12.03 The fundamental group: First properties 4.4 18.03 The fundamental group preserves products. Relative homotopies. Homotopy equivalence 4.5 19.03 The fundamental group preserves homotopy equivalences 5.1 02.04 Covering spaces: First definitions 5.2 08.04 Solutions to exercise sheet 5 5.3 09.04 Covering spaces: Path lifting property xxx 15.04 NO CLASS! xxx 16.04 NO CLASS! 5.4 22.04 Covering spaces: Homotopy lifting property. Lifting correspondence. Fundamental group of the circle. 5.5 23.04 Fundamental theorem of algebra. Review 5.6 29.05 Mock Exam END

## Exercise sheets

Exercise sheet 0 | Metric spaces|

Exercise sheet 1 | Topological spaces|

Exercise sheet 2 | Constructing topological spaces |

Exercise sheet 3| Topological properties: Compact spaces|

Typos in Exercise 3! iii) Needs the hypothesis X is locally compact. v) needs that X is compact.

Exercise sheet 4 | Connected and path connected topological spaces|

Exercise sheet 5 | Homotopy theory|

Mock Exam | Solutions|

## Exam

Solutions to the final exam :Solutions final exam

## Reference group

Rudolfs Sietinsons : rudolfs@stud.ntnu.no

Jacob Oliver Bruun : jacob.o.bruun@ntnu.no

Ingve Aleksander Hetland : ingveh@stud.ntnu.no

## References

We will not follow any particular textbook.

Some books on general topology:

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

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.