# MA3002 General Topology - Spring 2015

This is no longer a draft version of the course page. It will still be updated regularly.

### Messages

What's new?

- 17 Jul 2015: The re-exam will be oral.

What's old?

- 5 Jun 2015: Thanks to everyone who attended the course! It was a pleasure for me, and I hope that so it was for you as well. If you are thinking about doing a topology (or other) project with me, don't hesitate to contact me (or see my webpage first).

- 4 Jun 2015: The file exam2015.pdf has the exam problems with hints for solutions. You should have your results within a couple of days from now.

- 24 Apr 2015: I have corrected the setup for the problems on page 6.

- 21 Apr 2015: We will spend most of the remaining three lectures working on problems and exercises. I have added some more (page 5) to the file.

- 9 Apr 2015: I just learnt that Onsager Professor Xiang-Yu ZHOU will give his lecture
*Start with complex numbers*on Friday 10 Apr at 13:15 in KJL 2. I suggest that we go there after the break. The recap will continue on Tuesday 14 Apr if needed.

- 7 Apr 2015: Added a new reference and a link to old exams for MA3002.

- 27 Mar 2015: Next lecture on 10 April 2015 will be a recap. It's a good idea to review before and bring your questions to class.

- 24 Mar 2015: The lecture today had to be cancelled.

- 17 Mar 2015: Put the notes for the section on nets and filters into the references.

- 16 Mar 2015: Added some more exercises (on filters).

- 26 Feb 2015: Added some more exercises and improved the first worksheet some more.

- 6 Feb 2015: The worksheet now has hints for solutions on it. It also contains a remark that corrects a statement made during class.

- 5 Feb 2015: The file with the exercises keeps growing. I have added a worksheet on sums that we will work through tomorrow in class. Make sure that you bring enough coffee.

- 16 Jan 2015: Added more exercises and some
*challenges*. The latter are harder than what I would expect in an exam. But, some of you might want to try them anyway.

- 9 Jan 2015: Added exercises below.

- 6 Jan 2015: I have been in KJL24 to check: There are only twelve tables, but probably enough chairs to sit everyone.

- 5 Jan 2015: The first lecture will take place tomorrow as scheduled.

To bring everyone on the same page, I will discuss metric spaces and continuous maps between them.

### Course Information

### Lectures

- Tuesday, 14:15 - 16:00, KJL22
- Friday, 12:15 - 14:00, KJL24

### Lecturer

- Office hours: Tuesdays before the lecture in 1252, Sentralbygg II. Don't hesitate to come by and ask me.

### References

This is the official course text:

- J.R. Munkres. Topology. 2nd ed. Prentice Hall, 2000.

The following (parts of) books may also be useful for the purposes of the course.

- N. Bourbaki. Topologie générale. Hermann & Cie., 1940 and later. (in French)
- G.E. Bredon. Topology and geometry. Springer-Verlag, New York, 1993. (Chapter I and Appendix)
- T. tom Dieck. Algebraic topology. European Mathematical Society, Zürich, 2008. (Chapter 1)
- G. Laures, M. Szymik. Grundkurs Topologie. Spektrum Akademischer Verlag GmbH, 2009. (in German)
- L.A. Steen, J.A. Seebach. Counterexamples in Topology. Reprint of the second (1978) edition. Dover Publications, 1995.

- R. Ghrist. Elementary applied topology. 2014. www.math.upenn.edu/~ghrist/notes.html (Appendix: Background)

More references will be added here when they become relevant.

### Syllabus

- Part I General Topology in Munkres' book
- Chapter 1 reviews some set theory and logic
- Chapters 2, 3, 4 contain the core material
- Chapters 5, 6, 7, 8 cover additional topics

### (Preliminary) Lecture Plan

Week | Part | Topic | Reading |
---|---|---|---|

2.1 | I | Happy New Year and Welcome to (General) Topology! | Munkres, §20 |

2.2 | I | Metric spaces | Munkres, §21 |

3.1 | I | Topological spaces | Munkres, §12 |

3.2 | I | Continuous maps | Munkres, §18 |

4.1 | I | Bases | Munkres, §13 |

4.2 | I | Closed subsets | Munkres, §17 |

5.1 | II | Subspaces | Munkres, §16 |

5.2 | II | Quotients | Munkres, §22 |

6.1 | II | Products | Munkres, §15 |

6.2 | II | Sums | Worksheet |

7.1 | III | Connectivity | Munkres, §23 |

7.2 | III | Path connectivity | Munkres, §24 |

8.1 | III | Hausdorff spaces | Munkres, §31 |

8.2 | III | Separated maps | Worksheet |

9.1 | III | Compact spaces | Munkres, §26 |

9.2 | III | Proper maps | Munkres, §27 |

10.1 | III | Compact subspaces of R | Munkres, §19 |

10.2 | no lecture on March 6 | ||

11.1 | IV | Nets, filters, and convergence | Munkres, Supp. Ch. 3 |

11.1 | IV | Topology with filters | Notes |

12.1 | IV | Tychonoff's theorem | Munkres, §37 |

12.2 | IV | Examples | |

13.1 | cancelled | ||

13.2 | V | Function spaces | Munkres, §46 |

14 | no lectures in Week 14 | ||

15.1 | no lecture on Apr 7 | ||

15.2 | Recap | ||

16.1 | V | Local compactness | Munkres, §29 |

16.2 | V | Adjoint constructions | Munkres, §46 |

17.1 | Exercises, problems, questions, … | ||

17.2 | Exercises, problems, questions, … | ||

18.1 | Exercises, problems, questions, … | ||

18.2 | no lecture on May 1 |

### Exercises

Exercises will be done in class during the lecture time.

In addition, the file exercises.pdf, which will be updated regularly during the semester, also contains a list of exercises.

### Exam Info

- There will be a written examination on May 22, 2015.
- No printed or hand-written support material is allowed. A specific basic calculator is allowed.
- See here for more information.
- Old exams, as a 2.3 MB zip archive: old exams.zip

### Reference Group

- Marianne Stornes (BMAT)
- Marit Buset Langfeldt (MSMNFMA)
- Xenia Voellmy (IMEDIV/MN)