*Course Syllabus for*
# Topology

Topologi

## MATM16F, 7.5 credits

**Valid from:** Spring 2018

**Decided by:** Professor Thomas Johansson

**Date of establishment:** 2018-03-02

## General Information

**Division:** Mathematics

**Course type:** Course given jointly for second and third cycle

**The course is also given at second-cycle level with course code:** MATM16

**Teaching language:** English

## Aim

The aim of the course is to consolidate and generalize resultats which the student already has encountered in earlier courses in analysis, to equip him or her with an adequate language for higher studies in mathematics, and to develop his or her ability to work with abstract concepts, which are defined through of axioms. In particular the course should form a bridge between the mathematics courses at the Faculty of Engineering and more advanced mathematics courses.

## Goals

*Knowledge and Understanding*

For a passing grade the doctoral student must

- be able to define and explain basic concepts in general topology, such as topological space, connected space, continuous mapâ€¦
- be able to describe some common classes of av topologies
- be able to account for the different ways to define compactness.

*Competences and Skills*

For a passing grade the doctoral student must

- be able to prove the theorems in the course
- be able to determine if a given family of sets forms a topological space
- be able to prove the compactness of a given topological space
- be able to actively reason with use of the terms in the course.

## Course Contents

The fundamentals of the theory of metrical, topological and compact spaces. The Tietze extension theorem and Stone-Weierstrass approximation theorem. Elementary properties of Banach and Hilbert spaces.

## Course Literature

Munkres, J.: Topology. Pearson New International Edition. Pearson, 2017. ISBN 9780134689517.

Paperback of second edition from 2000.

**Types of instruction:** Lectures, seminars

**Examination formats:** Written exam, oral exam.
Compulsory assignments may occur,

**Grading scale:** Failed, pass

**Examiner:**

## Admission Details

**Admission requirements:** FMAA05 Calculus in one variable, FMAB20 Linear algebra, FMAB30 Calculus in several variables , FMAF01 Analytic Functions and FMAF05 Systems and Transforms.

## Course Occasion Information

**Course coordinators:**

**Web page:** http://www.ctr.maths.lu.se/course/top/