MATH3281

**Lecturer :** Dirk Schuetz

**Term :** Michaelmas 2012

- Monday 10:00 in E102
- Tuesday 15:00 in CG218

**Problems classes:**

- Friday 02.11 at 16:00 in CM107
- Monday 26.11 at 12:00 in CM221

**Literature**

- M.A.Armstrong,
*Basic Topology*. Undergraduate Texts in Mathematics, Springer-Verlag, New York-Berlin, 1983. - W.Fulton,
*Algebraic topology. A first course*. Graduate Texts in Mathematics, 153. Springer-Verlag, New York, 1995. - J.R.Munkres,
*Topology: a first course*. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1975. - W.S.Massey,
*Algebraic topology: an introduction*. Springer-Verlag, New York-Heidelberg, 1977.

**Assignments**

Homework | Date | Hand in | Solutions |

Problem Set 1 pdf | 16.10.12 | - | |

Problem Set 2 pdf | 29.10.12 | 06.11.12 | |

Problem Set 3 pdf | 12.11.12 | - | |

Problem Set 4 pdf | 26.11.12 | 04.12.12 | |

Problem Set 5 pdf | 10.12.12 | - |

**Lecture Outline**

Date | Outline |

11.12. | In this lecture we define real projective space and lens spaces, and show that SO(n) is connected. |

10.12. | In this lecture we discuss the action of the 3-sphere on 3-dimensional Euclidean space, orbits and orbit spaces. |

04.12. | In this lecture we show that O(n) is compact, define operations of topological groups on topological spaces, and discuss several such examples. |

03.12. | In this lecture we define topological groups and give plenty of examples. In particular, we show that the 3-sphere is a topological group. |

27.11. | In this lecture we have a closer look at the quotient topology, define identification maps and construct the Möbius band and the Klein bottle. |

26.11. | In this lecture we look at the Bolzano-Weierstrass Theorem and the Lebesgue Lemma, before defining quotient spaces and discussing some of their properties. |

20.11. | In this lecture we prove the Heine-Borel Theorem, thus giving a convenient criterion for compact subspaces in Euclidean space. We also discuss useful applications of it. |

19.11. | In this lecture we show that [0,1] is compact, and discuss all sorts of nice properties of compact spaces. |

13.11. | In this lecture we explain the difference between path-connectedness and connectedness, and introduce the concept of compactness. |

12.11. | In this lecture we look at more properties of connectedness, see more examples and compare it to path-connectedness. |

06.11. | In this lecture we give more examples of connected spaces, and look at some of their properties involving continuity. |

05.11. | In this lecture we have a closer look at some properties of the product topology, and consider the concept of connectedness. |

30.10. | In this lecture we define a basis of a topology, give equivalent notions of continuity using bases, closures etc, and look at product spaces. |

29.10. | In this lecture we deal with limit points, define the interior and the closure of a set, and see various examples. |

23.10. | In this lecture we define Hausdorff spaces, see that not every topological space is metric, and deal with continuity. |

22.10. | In this lecture we see examples of topological spaces, show how metric spaces fit into this concept, and look at closed sets. |

16.10. | In this lecture we relate open sets to continuity in metric spaces, and define topological spaces. |

15.10. | In this lecture we will define open and closed sets in metric spaces, and discuss some of their properties. |

09.10. | In this lecture we will look at convergence in metric spaces, define continous functions, and look at relations between them. |

08.10. | In this lecture we will define metric spaces, and give various examples of them. |

Last modified: 28.01.2013