# Matematik 3GT Fall 2004

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

### The latest news

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.

Matematik 5GE

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.

### What is topology?

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".

### Textbook

Our textbook

James R. Munkres, Topology. Second edition. Prentice-Hall

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.

### Course Plan

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.

### Problem sessions

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.

### Exam

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.