Topology III

MATH3281

Lecturer : Dirk Schuetz

Term : Michaelmas 2014

Lectures :
  • Tuesday 13:00 in PH8
  • Friday 17:00 in CM101

Problems classes:

  • Monday 27.10, 10.11, 24.11 and 08.12 at 12:00 in PH30

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. Here are some notes on what has been covered this term. Notice that examples are rather short and proofs are omitted, so you should still get your notes from the lectures. Here are some notes on Quaternions which give a bit more detail why the 3-sphere and SO(3) are closely related.

Assignments

Homework Date Hand in Solutions
Problem Set 1 pdf 10.10.14 24.10.14 pdf
Problem Set 2 pdf 24.10.14 07.11.14 pdf
Problem Set 3 pdf 07.11.14 21.11.14 pdf
Problem Set 4 pdf 21.11.14 05.12.14 pdf

Problem Class Date Solutions
pdf 27.10.14 pdf
pdf 10.11.14 pdf
pdf 24.11.14 pdf
pdf 08.12.14 pdf

Lecture Outline

Date Outline
12.12. In this lecture we show that SO(n) is connected.
09.12. In this lecture we see how the 3-sphere is a subgroup of SO(4) and construct an interesting homomorphism to SO(3). We also define orbit spaces of group actions.
05.12. In this lecture we consider left and right translation in a topological group, see that O(n) is compact, and define actions of topological groups on topological spaces.
02.12. In this lecture we define topological groups, and see plenty of examples.
28.11. In this lecture we define identification maps, and show how they help us to understand quotient spaces.
25.11. In this lecture we prove the Bolzano-Weierstrass for general compact spaces and meet the Lebesgue Lemma. We also introduce the quotient topology.
21.11. In this lecture we show that the product of compact spaces is compact, and we prove the Heine-Borel Theorem.
18.11. In this lecture we show that [0,1] is compact, and also see more properties of compact spaces.
14.11. In this lecture we define path connected spaces, see their relation to connectedness, and define compactness.
11.11. In this lecture we will see more about connected spaces, connected components, and see some examples.
07.11. In this lecture we will see more about connected spaces, in particular interaction with continuous functions and subsets.
04.11. In this lecture we learn more about the product topology, introduce the concept of connectedness, and show that the reals are connected.
31.10. In this lecture we see more relations between a basis and a topology, and define the product topology on a cartesian product of topological spaces.
28.10. In this lecture we define the interior and closure of sets, relate them to limit points, and define a basis for a topology.
24.10. In this lecture we will see how we can build interesting topological spaces from continuous functions, and meet the concept of homeomorphism, neighborhood and limit point.
21.10. In this lecture we will look at a few more examples of topological spaces, define closed sets, Hausdorff spaces and continuous functions between topological spaces.
17.10. In this lecture we express continuity in terms of open sets, define topological spaces and see various examples.
14.10. In this lecture we define open and closed ball, and use them to define open and closed sets in metric spaces.
10.10. In this lecture we discuss continuity of functions between metric spaces, and how this gets detected by sequences.
07.10. In this lecture we will define metric spaces, and give various examples of them.

Last modified: 12.12.2014