Study information

Topology and Metric Spaces - 2024 entry

MODULE TITLETopology and Metric Spaces CREDIT VALUE15
MODULE CODEMTH3040 MODULE CONVENERProf Nigel Byott (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 0 11 weeks 0
Number of Students Taking Module (anticipated) 24
DESCRIPTION - summary of the module content

Topology and metric spaces provide a set of powerful tools that are used in many other branches of mathematics (from Algebraic Topology and Numerical Analysis to Dynamical Systems and Ergodic Theory). Fundamental to these topics is the idea of generalising the idea of “closeness” of two objects in a set to a very general setting. These techniques are fundamental to the understanding of more advanced topics in mathematics such as Measure Theory, Functional Analysis, Algebraic Topology and Algebraic Geometry.

This course aims to give an introduction to topology and metric spaces. In every section covered in this course we will start by studying the definitions and then we will present examples and some basic properties. Some important theorems will be stated and proved. With this module you will have the opportunity to further refine your skills in problem-solving, axiomatic reasoning and the formulation of mathematical proofs.

Pre-requisite - MTH2001 or MTH2008

AIMS - intentions of the module

The objective of this module is to provide you an introduction to Topology and Metric Spaces. Our main objective will be to define the basic concepts clearly and to provide proofs of useful theorems.

INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)

On successful completion of this module you should be able to:

Module Specific Skills and Knowledge

1. Recall and apply key definitions in Analysis;

2. State, prove and apply core theorems in Topology and metric spaces.

Discipline Specific Skills and Knowledge

3. Extract abstract problems from a diverse range of problems;

4. Use abstract reasoning to solve a range of problems.

Personal and Key Transferable / Employment Skills and Knowledge

5. Think analytically and use logical argument and deduction;

6. Communicate results in a clear, correct and coherent manner.

SYLLABUS PLAN - summary of the structure and academic content of the module

- Review of some real analysis: Real numbers, real sequences, limits of functions, continuity, intervals, set theory. (3 lectures) 
 
- Metric spaces: Definition and examples, open and closed sets in metric spaces, equivalent metrics, examples. (6 lectures) 
 

- Complete metric spaces: Definition and examples, Fixed point theorems, the contraction mapping theorem. (3 lectures). 
 
- Topological spaces: Bases, sub-bases and weak topologies, topologies of subspaces and products, homeomorphisms. (6 lectures) 
 
- The Hausdorff condition: separation axioms, Hausdorff space, regular topological space. (3 lectures) 
 
- Compact spaces: Definition, Compactness of [a,b], properties of compact spaces, continuous maps on compact spaces. An inverse function theorem. (5 lectures) 
 
- Connected spaces: Connectedness, components, path-connectedness. (4 lectures) 
 
- Revision (3 lectures) 

 

LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
Scheduled Learning & Teaching Activities 33 Guided Independent Study 127 Placement / Study Abroad 0
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
Category Hours of study time Description
Scheduled learning and teaching activities 33 Lectures

Example classes

Guided Independent Study

 127
Example classes
 
Studying additional recordings complementing lectures, and reading material, examples sheets and revision

 

ASSESSMENT
FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade
Form of Assessment Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Coursework problem sheets 10 hours 1-6 Problems discussed in lectures used as problems class. Written comments on scripts available on request. 
       
       
       
       

 

SUMMATIVE ASSESSMENT (% of credit)
Coursework 20 Written Exams 80 Practical Exams 0
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Coursework 1– based on questions submitted for assessment 10 10 hours 1-6 Annotated script and written/verbal feedback
Coursework 2– based on questions submitted for assessment 10 10 hours 1-6 Annotated script and written/verbal feedback
Written Exam- closed book 80 2 hours  1-6 Written/verbal on request, SRS
         
         
         

 

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)
Original Form of Assessment Form of Re-assessment ILOs Re-assessed Time Scale for Re-assessment
Written Exam  Written Examination (2 hours)  1-6 Referral/Deferral Period
Coursework 1  Coursework 1 1-6 Referral/Deferral Period
Coursework 2  Coursework 2 1-6 Referral/Deferral Period

 

 

 

RE-ASSESSMENT NOTES

Deferrals: Reassessment will be by coursework and/or written exam in the deferred element only. For deferred candidates, the module mark will be uncapped.

Referrals: Reassessment will be by a single written exam worth 100% of the module only. As it is a referral, the mark will be capped at 40%.

RESOURCES
INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener

Basic reading:

 

ELE: http://vle.exeter.ac.uk/

 

Web based and Electronic Resources:

 

Other Resources:

 

Reading list for this module:

Type Author Title Edition Publisher Year ISBN
Set Falconer, K. Fractal Geometry 2nd edition Wiley 2003 978-0470848623
Set Sutherland, W.A. Introduction to Metric and Topological Spaces Oxford Science Publications
Set Rudin, W. Real and Complex Analysis Third McGraw Hill 1987 978-0070619876
Set Charles Chapman Pugh Real Mathematical Analysis Undergraduate Texts in Mathematics, Springer
Set James R Munkres Topology Prentice Hall
CREDIT VALUE 15 ECTS VALUE 7.5
PRE-REQUISITE MODULES MTH2001, MTH2008
CO-REQUISITE MODULES
NQF LEVEL (FHEQ) 6 AVAILABLE AS DISTANCE LEARNING No
ORIGIN DATE Tuesday 12th March 2024 LAST REVISION DATE Monday 18th March 2024
KEY WORDS SEARCH None Defined

Please note that all modules are subject to change, please get in touch if you have any questions about this module.