# Mathematics

## MTH3040 - Topology and Metric Spaces (2019)

MODULE TITLE CREDIT VALUE Topology and Metric Spaces 15 MTH3040 Dr Ana Rodrigues (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 as well as applications to basic concepts of measure theory. 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 Analysis

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. (4 lectures)

- Topological spaces: Bases, sub-bases and weak topologies, topologies of subspaces and products, homeomorphisms. (4 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. (3 lectures)

- Connected spaces: Connectedness, components, path-connectedness. (3 lectures)

- Complete metric spaces: Definition and examples, Fixed point theorems, the contraction mapping theorem. (4 lectures)

- Introduction to measure theory: Measure of plane sets. Outer and inner measure of a set. Measurable set (in the sense of Lebesgue). Some fundamental properties of Lebesgue measure and measurable sets. Definition and fundamental properties of measurable functions. (3 lectures)

- If time allows, a selection from the following: σ-algebras. Positive Borel measures. The Riesz representation theorem. Lp-spaces. Elementary Hilbert space theory. Banach spaces. Baire’s theorem. (3 lectures)

- Revision (3 lectures)

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

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 All Written comments on scripts

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams Practical Exams 20 80 0
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Coursework – based on questions submitted for assessment 20 2 assignments, 30 hours total All Annotated script and written/verbal feedback
Written Exam – closed book 80 2 hours All 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
All Above Written Examination All August Ref/Def Period

RE-ASSESSMENT NOTES

If a module is normally assessed entirely by coursework, all referred/deferred assessments will normally be by assignment.

If a module is normally assessed by examination or examination plus coursework, referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 40% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

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

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

Web based and Electronic Resources:

Other Resources: