Practical information from SIS. The semester. The course will be taught in English.

Monday Jan 10: What results can I freely refer to at the exam? See E-mail from Esben.

Monday Jan 10 at 12.15 in Aud 10: At this last session with your instructor you can ask all sorts of questions relevant to the curriculum. (Spørgetime!)

Hold 2: To make up for the cancelled session, Hold 2 will meet Friday 03.12 12-14 in A104.

Wednesday November 3rd, hold 3 will meet in A110

Monday November 1st, hold 2 will meet in A104 (all other Mondays in A106)

THE CONTINUUM IN PHILOSOPHY AND MATHEMATICS, November 25-27, Copenhagen.

Why don't you ask a topologist? Or look at the Topology Atlas? Here is an excellent introduction to topology with several pictures and animantions.

The short answer is: Topology is the study of continuity. If you want a more elaborate answer, you can see here what the topologists themselves think topology is or consult The Mathematical Atlas for General Topology. See the essay on the History of Topology if you want to know where it all came from. See also the links below.

The word "topology" comes from the old Greek words "topos", meaning "place", and "logos", meaning "word".

Our textbook

is available at Universitetsbogladen.

The incomplete Dictionary (contributions are welcome) provides translation of topological terms into a few European languages.

My own lecture notes (dvi) (pdf) for the course.

Those of you who prefer something shorter, may find Chapter I: General Topology from G.E. Bredon: Topology and Geometry a useful synopsis. Also Chapter 2 of J.M. Lee: Introduction to Topological Manifolds can be recommended. See also below for more relevant literature.

Week | Chapters | Subjects | Exercises | Solutions |
---|---|---|---|---|

1 | §1 - §4 | Sets, operations with sets, functions, relations Slides (dvi) (pdf) |
§1: 2, 4, 7 §2: 1, 2, 3, 4, 5, 6 |
Euler walks (pdf) |

2 | §5 - §7 | Finite, countable and uncountable sets, cardinal
numbers. Slides (dvi) (pdf) |
§3: 2, 4, 7, 11, 12, 13, 15 §4: 3, 4, 5 |
Exercises §3 (dvi) (pdf)
Exercises §4 (dvi) (pdf) |

3 | §9 - §11 | Infinite sets, well-ordered sets, ordinal
numbers, Axiom of
choice,
Zermelo's well ordering theorem,
Hausdorff's maximum principle,
Zorn's lemma. W.H. Woodin, The Continuum Hypothesis I and II |
§5: 3 §6: 3, 4§7: 1, 3, 5, 6. |
Exercises §6 (dvi) (pdf) §7#6 Exercises §10 (dvi) (pdf) |

4 | §12 - §16, §19 | Topological spaces, basis for a topology, order topology, product topology, subspace topology. | §9: 5, 6, 7 §10: 1, 2, 3, 5, 7 §11: 8 (p. 72) |
Relations between topological spaces (dvi) (pdf) Exercises §11 (dvi) (pdf) Exercises §13 (dvi) (pdf) |

5 | §16 - §18 | Closed sets, continuous functions, open and closed functions. | §13: 1, 4, 5, 6, 7, 8.§16: 1, 3 |
Exercises §16 (dvi) (pdf) Exercises §17 (dvi) (pdf) Exercise 17.19 (dvi) (pdf) Exercises §18 (dvi) (pdf) |

6 | §19, §22 | Quotient topology, Hausdorff spaces, product topology. (I will be in Barcelona to give a talk - Ryszard Nest lectures instead of me!) |
§16: 4, 6, 9, 10§17: 3, 6, 8, 9, 13, 14. |
Exercises §19 (dvi) (pdf) Exercises §22 (dvi) (pdf) |

No lecture | No problem session | Fall Break | ||

7 | §20 - §21 §23 - §24 |
Metric topology. Connected and path connected spaces. |
§17: 10, 11, 12, 19§18: 1, 2, 7, 8, 13 §31: 5 §19: 10 |
Exercises §20 (dvi) (pdf) Exercises §23 (dvi) (pdf) Exercises §24 (dvi) (pdf) |

8 | § 25 | Components and path components. Locally connected and locally path connected spaces. |
§19: 7 §20: 5 §22: 2, 3, 5 §23: 2, 4, 5, 6, 11 |
Exercises §25 (dvi) (pdf) |

9 | §26 - §27 | Compactness. Compact subsets of linearly ordered spaces. Compact subsets of Euclidean spaces (Heine-Borel theorem). Alexander's horned sphere |
§24; 1, 2, 8, 10, 11§25: 1, 2, 3, 4 |
Exercises §26 (dvi) (pdf) Exercises §27 (dvi) (pdf) |

10 | §28 - §29 | Extreme value theorem,
Lebesgue lemma, Uniform continuity
theorem. Limit point and sequential compactness. Local compactness. Alexandroff compactification. |
§26: 1, 2, 3, 5, 7, 12§27: 1, 3, 6 |
Exercises §28 (dvi) (pdf) Exercises §29 (dvi) (pdf) |

11 | §30 - §31 | Countability axioms.
Lindelöf spaces. Separation axioms. |
§28: 1, 6 §29: 1, 3, 5, 6, 7, 11 |
Exercises §30 (dvi) (pdf) Exercises §31 (dvi) (pdf) |

12 | §32 - §33 §35 |
Normal spaces. Urysohn lemma, Tietze extension theorem. |
§30: 3, 7 §31: 1, 2, 3, 6, 7 |
Exercises §32 (dvi) (pdf) Exercises §33 (dvi) (pdf) |

13 | §34, §37 |
Tychonoff theorem. The diagonal embedding theorem. Second countable regular spaces. Urysohn metrization theorem. |
§32: 1, 3, 5, 6 §33: 4, 5, 7 §29: 10, §30: 5,
6 |
Exercises §34 (dvi) (pdf) Exercises §35 (dvi) (pdf) |

14 | §36, §38 | Completely regular spaces Stone- Cech compactification Embeddings of manifolds John Milnor:Towards the Poincaré conjecture |
§34: 3, 4 §35: 4, 5, 6, 7, 9§36: 1 |
Exercises §36 (dvi) (pdf) |

15 | §1 - §38 | Review and sins of omission Manifolds and surfaces |
Exam January 2004 (dvi) (pdf) Exam June 2004 (dvi) (pdf) |
Exercises §38 (dvi) (pdf) |

There is no teaching in week 42, Oct 11-15.

The problem sessions will start the **second** week
of the semester. Your instructors are:

Morten Poulsen Hold 1,
Mondays 10-12, N030

Anders Storm Hansen Hold
2, Mondays 12-14, A106

Esben Bistrup
Halvorsen Hold 3, Wednesdays 12-14, N004

Exercises written in boldface, such as §29:
**11**, are written assignments.

The three hour written exam is scheduled to January 14th 2005.
Books, notes, and calculators are allowed during the exam. You may
write your answers in pencil. The exam will cover §§ 1 -
38 (including all worked exercises). In your answers to the exam
problems you may freely refer to anything in Munkres' book or in my
lecture notes.

- Solution (dvi ) (pdf) to the August 2005 exam (dvi) (pdf)
- Solution (dvi ) (pdf) to the January 2005 exam (dvi) (pdf)
- Solution (dvi ) (pdf) to the June 2004 exam (dvi) (pdf)
- Solution (dvi ) (pdf) to the January 2004 exam (dvi) (pdf)
- Solution (dvi ) (pdf) to the June 2003 exam (dvi) (pdf)
- Solution (dvi ) (pdf) to the January 2003 exam (dvi) (pdf)
- 3GT Exams before 2003

- R. Engelking: General Topology.
- R. Engelking, K. Sieklucki: Topology. A geometric approach.
- J. Dugundji: Topology.
- J. L. Kelley: General Topology.
- A. Levy: Basic Set Theory
- T. Moore: Elementary general topology.
- John M. Lee, Introduction to Topological Manifolds.
- Dennis Roseman, Elementary Topology. Prentice Hall 1999.
- W.A. Sutherland, Introduction to Metric and Topological Spaces. Oxford Science Publications 1975. (Several reprintings)
- Stephan C. Carlson, Topology of Surfaces, Knots, and Manifolds. A first undergraduate course. John Wiley 2001.
- D.W. Farmer and T.B. Stanford, Knots and Surfaces: A Guide to Discovering Mathematics. American Mathematical Society 1996.
- J. Stillwell, Classical Topology and Combinatorial Group Theory. GTM 72. Springer Verlag.
- P.A. Firby and C.F. Gardner, Surface Topology. Ellis Horwood 1982.
- Richard M. Crownover, Introduction to Fractals and Chaos. Jones and Bartlett 1995.
- I. Adamson: A general topology workbook. (Contains many worked solutions to exercises.)

Topology Atlas History of topology The Geometry Junkyard The Knot Plot Site Toplogy with Maple |
Algebraic and
Geometric Topology On-line topology courses C. Berg: Topologi (In Danish) |
The Topological
Zoo Klein Bottles for sale. Almen Matematisk Dannelse (dvi)
(pdf)
(in Danish)Classification of covering maps From singular chains to Alexander homology |

Back to Jesper's home page.

Jesper Michael Møller Last modified: Mon Jun 18 17:19:25 CEST 2007