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
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 main properties of subsets of topological spaces including criteria for the set to be open, closed, compact, connected, path connected
- Understand the behaviour and properties of sequences in general metric spaces, including convergence, and the Cauchy property.
- 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.
- 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.
Metrics on a set:
- Definitions and main examples
oMetrics on Rn
oNormed vector spaces as metric spaces
oSpaces of sequences
oSpaces of continuous functions
- 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
- 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
|Completion of assessment task||20|
|Preparation for scheduled sessions||12|
|Supervised time in studio/workshop||6|
|Wider reading or practice||10|
|Total study time||150|
Resources & Reading list
W.A. Sutherland. Introduction to Metric and Topological Spaces.
This is how we’ll formally assess what you have learned in this module.
This is how we’ll assess you if you don’t meet the criteria to pass this module.
Repeat type: Internal & External