## Introduction to Topology

### Course number: 201-10091; Spring Semester 2014

#### Ben Gurion University of the NegevDepartment of Mathematics

Dr. Assaf Hasson
Office No. 204, Bldg 58.
Office hours: Sundays and Tuesdays 11:00-12:00.

Class hours:
Sunday 13h-15h, Bldg 32 class No. 309
Tuesday 12h-14h, Bldg 72 class No. 217

There will be 7 homework assignements, gradually published on this web page. You will have to pass at least 5 of them in order to attend the final exam. You can earn 10 points by grading handouts to one work sheet, and writing its solutions. Both grading and homework submission will be done in pairs.

The take home exam : to be electronically submitted not after Oct. 20th.

#### Syllabus and lecture notes

The course is intended to introduce the students to the basic concepts of point-set topology. If time allows, we will try to give some flavour of algebraic topology as well. We hope to cover:
• Introduction: basic definitions and examples, metric spaces, construction of topological spaces, connectedness and path connectedness, notions of convergence, Baire's theorem.
• Tychonov's Theorem: The product topology, compactness, filters and ultra-filters and their applications, Tychonov's theorem.
• Metrisability: Axioms of countability and separation, Urysohn's lemma, Urysohn's metrisability theorem. Some useful examples. The Stone-Cech compactification and some applications.
• The fundamental group of the circle: Homotopy, the fundamental group, simply connected spaces, covering spaces, computing the fundamental group, Van-Kamppen's theorem (special case). Applications (as time allows): Brouwer's fixed point theorem in the plane, the fundamental group of the sphere, Jordan's theorem in the plane.
Important notice: though the lecture notes above cover practically everything we will do in class, they were written for my personal use, and therefore may contain typos and even mistakes. This is not, by any means, the "official text" of the course.

#### RECOMMENDED BIBLIOGRAPHY:

The couse will not follow closely any textbook. Good references for almost all we will do are:
• Munkres, Topology (A first course).
• Kelley, General topology.
Though Kelley's book does not cover the last part of the course.

#### Exercises

Homework assignments, as well as homework grades, will be published on the web-page.

A recap exercise (and its solution) on point-set topology to help you sum up what we did in class in that part of the course. This assignment is not for submission.

All exercises should be submitted in class on Tuesdays. Any other form of submission will not be accepted.

 Exercise 1 Solution 1 Exercise 2 Solution 2 Exercise 3 Solution 3 Exercise 4 Solution 4 Exercise 5 Solution 5 Exercise 6 Solution 6 Exercise 7 Solution 7