This is the web-page for the course Topologie en Meetkunde, 2007. The web-page for the previous year (including lecture notes FOR THAT YEAR, exercises, etc) can be found here.

Topologies. These are lecture notes for the first 3-4 lectures. Here are a few more things discussed in the class in these lectures.

Here is a set of more exercises.

Reminder on group theory.

Seifert-van Kampen theorem.

Computations of fundamental groups.

Here you can see the exercises for Exam B of the previous year.

This is the web-page for the course Topologie en Meetkunde, 2007. The web-page for the previous year (including lecture notes FOR THAT YEAR, exercises, etc) can be found here. General presentation:

Given a set of points X, a topology on X is some "extra-data" on X which allows us to make sense of statement such as:

- two points of X are "close to each other"

- a sequence of points (in X) converges to another point.

- a function f: X --> |R is continuous

or other "topological statements" which you might have seen in analysis courses (such as compactness, connectedness).

Familiar sets of points such as the circle, the sphere, the torus, although they are all "isomorphic" as sets (i.e. there are bijections between them), our intuition tells us that they look quite different. That is because our intuition sees not only the bare sets of points, but also the natural topologies on them (the meaning of "a point gets closer to another one" beeing clear in each of these examples). And our intuition is right: the circle, the sphere and the torus are not isomorphic as topological spaces.

Apart from "geometric examples" of topological spaces (circle, tory, cube, etc). you have probably seen (or you will see) other interesting examples coming from analysis, functional analysis, distribution theory, etc. Such as: the space of bounded continuous functions, the space of compactly supported smooth functions, the space of distributions on an open set in the euclidean space.

Question A: A central question in topology is: given two topological spaces X and Y, how do we decide if they are isomorphic as topological spaces? Example: how do you PROVE that the circle and the sphere are not? Or that [0, 1] and [0, 1) are not?

Question B: Another important question is: given two spaces X and Y, when can one obtain one from the other by "pushing" and "pulling" them, without "breaking them". Particular case: given a space X and a subspace A, when can one "push X inside A", without "breaking it", and without leaving X. Question B is often very useful for answering Question A, and it is at the origin of "Algebraic Topology".

** The aim** of the course is to make the students know, understand, and feel what topological spaces are, give them the tools to answer questions as the two posed above and, in particular, discuss in detail the case of surfaces.

** The content** of the course is as follows. The first few courses are dedicated to the notion of topology and immediate topological notions, to examples and to constructions which allow us to produce new examples out of old ones (direct product, direct sum, quotient topology, etc). Then we concentrate on deeper topological properties such as compactness, local compactness, connectedness, normality, which allow us to attack Question A in many examples. In the second part we discuss more refined tools for answering the two questions mentioned above and for understanding the classification of surfaces: the fundamental group, computations using covering spaces, computations using the Seifert-van Kampen theorem and cell-attachements.

More info about the course:

The course is given every Monday, from 13:15 to 15:00, in Wisk K11. Starting date: February 5, 2007. Lecturer: Marius Crainic.

(Het college wordt gegeven op maandagmiddag van 13.15 tot 15.00 door dr. M. Crainic in Wisk K11.)

The exercise classes will take place every Thursday. In charge: Camilo Arias Abad.

(Het practicum is op donderdag van 13.15 tot 15.00, in MG 012. De practicumleider is Camilo Arias Abad.)

Text book: "Topology" by James A. Munkres, 2-nd edition. The text book will be used only ocasionally. During the course, we will make available lecture notes. The lecture notes from the last year can be seen here.

Rules for the examniation: there are two exams, A and B. Both of them have to be passed, with a minimum mark of 5 for each one of them, and with an average of at least 6. One part of the exam is not valid for more then a year. There will be one chance to repeat the exam.

(Tentamenregeling: er zijn twee deeltentamens, A en B. Deze hebben geen zelfstandige betekenis: om het vak te halen moet over beide tentamens een gemiddelde van minstens 6 worden gehaald; met dien verstande dat de uitslag van beide tentamens minstens 5 moet zijn. Een deeltentamen blijft niet geldig tot volgend jaar. Er is een herkansing over de gehele stof.)