# TMA4190 Introduction to Topology - Spring 2023

Schedule | Room | ||
---|---|---|---|

Lectures: | Tuesday | 12.15 - 14.00 | KJL2 |

Friday | 14.15 - 16.00 | KJL2 |

Instructor: | Fernando Abellán | ||
---|---|---|---|

Webpage: | https://www.ntnu.no/ansatte/fernando.a.garcia | ||

Office: | 1204 Sentralbygg 2 | ||

Email: | fernando [dot] a [dot] garcia [at] ntnu [dot] no |

## About this course

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 10 and 13.

Lecture | Date | Topic |
---|---|---|

0.1 | 10.01 | Introduction. Metric spaces and continuous functions. |

0.2 | 13.01 | Characterization of continuous functions of metric spaces |

1.1 | 17.01 | Topological spaces and continuous maps |

1.2 | 20.01 | Solutions to Exercise sheet 0 |

1.3 | 23.01 | Closed sets, continuity in terms of closed sets, closure of a set |

1.4 | 27.01 | Continuity in terms of closure. Basis of a topology |

2.1 | 31.01 | Constructing topological spaces: Subspace topology and Cartesian product |

2.2 | 03.02 | Solutions to Exercise sheet 1 |

2.3 | 07.02 | Universal property of the Cartesian product. Quotient spaces |

2.4 | 10.02 | Open and closed maps. |

3.1 | 14.02 | Countability axioms |

3.2 | 17.02 | Separability axioms: Hausdorff spaces . Connected spaces |

3.3 | 20.02 | Connected spaces: The real numbers are connected. |

3.4 | 24.02 | Solutions to Exercise sheet 2 |

3.5 | 28.02 | Compact spaces |

3.6 | 03.03 | Compact spaces: Cartesian product of compact spaces is compact. |

3.7 | 07.03 | One point compactification and examples. |

3.8 | 10.03 | Solutions to exercise sheet 3. |

4.1 | 14.03 | Homotopy classes of maps. Relative homotopies. |

4.2 | 17.03 | Products of paths. The fundamental group. |

4.3 | 21.03 | Properties of the fundamental group: Functoriality. |

4.4 | 24.03 | No class! |

4.5 | 28.03 | Homotopy equivalences and the fundamental group |

4.6 | 31.03 | Solutions to exercise sheet 4. |

5.1 | 10.04 | Scheme of lecture |

5.2 | 13.04 | Computing the fundamental group of the circle |

5.3 | 18.04 | The fundamental theorem of algebra via homotopy theory. |

6 | 23.04 | Final lecture: Mock exam |

## Exercise sheets

Exercise sheet 0: Metric spaces Updated 18.01

Exercise sheet 0: Solutions to Exercise 7

Exercise sheet 1: Topological spaces

Exercise sheet 2: Constructing topological spaces

Exercise sheet 3: Connected and compact topological spaces

Exercise sheet 4: Homotopy theory

## 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.