Topology III

MATH3281

Lecturer : Dirk Schuetz

Term : Michaelmas 2012

Lectures :
  • 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

The following is a list of books on which the lecture is based. Although we will not follow a book strictly, the material can be found in them and they may sometimes offer a different approach.
  • 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.

Handouts

Here is a handout on Sets and Functions, which lists properties of images and pre-images.

Assignments

Homework Date Hand in Solutions
Problem Set 1 pdf 16.10.12 - pdf
Problem Set 2 pdf 29.10.12 06.11.12 pdf
Problem Set 3 pdf 12.11.12 - pdf
Problem Set 4 pdf 26.11.12 04.12.12 pdf
Problem Set 5 pdf 10.12.12 - pdf

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