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# 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

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#### Learning Outcomes

##### Learning Outcomes

Having successfully completed this module you will be able to:

• Should be able to check if a given real valued function of two variables is a metric on a set
• Should be able to distinguish differences and similarities between main types of geometries
• Should be able to check if a given sequence in metric space is Cauchy
• Should be able to check if a given family of subsets of a set is a topology
• Should be able to decide if a function between topological spaces is continuous
• Should be able to decide if a function of metric spaces satisfies the Lipschitz condition or is a contraction
• Should be able to apply the Contraction Mapping theorem to Fixed Point Problems
• Should be able to use appropriate theorems to decide if a subset of a topological space satisfies one of the properties like being closed, connected, path connected or compact.

### 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 • Spherical geometry Properties of metric spaces: • Basic shapes: open balls • 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
Tutorial12
Wider reading or practice10
Lecture36
Supervised time in studio/workshop6
Completion of assessment task20
Preparation for scheduled sessions12
Follow-up work30
Revision24
Total study time150

#### Resources & Reading list

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

### Assessment

#### Summative

MethodPercentage contribution
Class Test 20%
Written exam  (2 hours) 80%

#### Referral

MethodPercentage contribution
Written exam 100%

#### Repeat Information

Repeat type: Internal & External

### Linked modules

Pre-requisite: MATH2039

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