The University of Southampton
Courses

# MATH2049 Geometry and Topology

## Module Overview

Geometry has grown out of efforts to understand the world around us, and has been a central part of mathematics from the ancient times to the present. Topology has been designed to describe, quantify, and compare shapes of complex objects. Together, geometry and topology provide a very powerful set of mathematical tools that is of great importance in mathematics and its applications. This module will introduce the students to the mathematical foundation of modern geometry based on the notion of distance. We will study metric spaces and their transformations. Through examples, we will demonstrate how a choice of distance determines shapes, and will discuss the main types of geometries. An important part of the course will be the study of continuous maps of spaces. A proper context for the general discussion of continuity is the topological space, and the students will be guided through the foundations of topology. Geometry and topology are actively researched by mathematicians and we shall indicate the most exciting areas for further study. One of the pre-requisites for MATH3076, MATH3080, MATH3086, MATH6079, MATH6094, MATH6095, MATH6137 and MATH6138

### Aims and Objectives

#### Learning Outcomes

##### Learning Outcomes

Having successfully completed this module you will be able to:

• Understand properties of metrics on a wide range of spaces and be able to check if a given real valued function of two variables is a metric on a set.
• Understand the behaviour and properties of sequences in general metric spaces, including convergence, and the Cauchy property.
• Understand the Contraction Mapping Theorem and use it to solve a variety of fixed point problems including proving the existence of solutions of differential equations.
• Understand the main properties of subsets of topological spaces including criteria for the set to be open, closed, compact, connected, path connected
• Be familiar with the notion of continuity of functions between metric spaces condition, and general topological spaces, including isometries, Lipschitz maps, homeomorphisms and quotient maps.

### Syllabus

Metrics on a set: • Definitions and main examples o Metrics on Rn o Normed vector spaces as metric spaces o Spaces of sequences o Spaces of continuous functions o Length spaces Geometry: • Isometries • Euclidean geometry and classification of Euclidean isometries • Further examples of geometries Properties of metric spaces: • Sequences and convergence, completeness • The Bolzano-Weierstrass theorem • Continuous functions of metric spaces • Lipschitz functions and contractions • Uniform and pointwise convergence in spaces of continuous functions of real variable • The Contraction Mapping Theorem • Picard’s Theorem and differential equations Topology: • Definition of topology, basis of topology • Metric topology, open and closed sets in metric spaces • Interior and closure • Continuous functions and homeomorphisms • Connected and path connected sets • Compact sets • Tychonoff’s theorem and its consequences, including the Heine-Borel theorem • Outline of quotient topology and basic quotient constructions.

### Learning and Teaching

#### Teaching and learning methods

Lectures, problem classes, workshops, private study

TypeHours
Follow-up work30
Tutorial12
Preparation for scheduled sessions12
Revision24
Lecture36
Supervised time in studio/workshop6
Total study time150

W.A. Sutherland. Introduction to Metric and Topological Spaces.

### Assessment

#### Summative

MethodPercentage contribution
Coursework 40%
Written assessment 60%

#### Referral

MethodPercentage contribution
Written assessment 100%

#### Repeat Information

Repeat type: Internal & External