# MAS8706 : Metric Spaces and Topology

• Offered for Year: 2024/25
• Module Leader(s): Dr Christian Bönicke
• Owning School: Mathematics, Statistics and Physics
• Teaching Location: Newcastle City Campus
##### Semesters

Your programme is made up of credits, the total differs on programme to programme.

 Semester 1 Credit Value: 10 ECTS Credits: 5.0 European Credit Transfer System

#### Aims

To introduce the language of metrics, a formalisation of the intuitive notion of distance. To illustrate how the framework of metric and topological spaces makes central concepts from real analysis, such as convergence of sequences and continuity of functions, applicable to a vast number of different settings.

Module Summary

The notion of distance between two objects is ubiquitous throughout mathematics and the sciences. From the familiar distance between two points in Euclidean space to the length of a shortest path between two computers in a network, there are various ways of making sense of the concept of distance depending on the context. This module presents the basic theory of metric spaces, that is, sets equipped with an abstract notion of distance. This approach leads to a widely applicable theory, which is why metric spaces form a foundational part of the modern mathematics curriculum.

The module will discuss basic analytic notions (convergence, Cauchy sequences, continuity) that will put the familiar theory from real analysis into a broader context, but also introduce some new topological and geometric ideas (connectedness, compactness, isometries, contractions).

#### Outline Of Syllabus

Metric spaces, subspaces, product spaces. Convergence of sequences. Cauchy sequences, complete metric spaces. Functions between metric spaces (continuous functions, isometries, contractions), the Banach Fixed-Point Theorem. Open sets and closed sets, convergence and continuity in terms of open sets. Interior, closure, boundary. Compact spaces, the Heine-Borel Theorem. Connected metric spaces. Topological spaces.

#### Teaching Methods

##### Teaching Activities
Category Activity Number Length Student Hours Comment
Structured Guided LearningLecture materials101:3015:00Lecture videos and accompanying lecture notes, completion of worksheet/NUMBAS quiz
Scheduled Learning And Teaching ActivitiesLecture21:002:00Revision Lectures
Scheduled Learning And Teaching ActivitiesLecture51:005:00Plenum sessions where final questions on course material can be addressed
Guided Independent StudyAssessment preparation and completion101:0010:00Completion of in course assessments
Scheduled Learning And Teaching ActivitiesWorkshops201:0020:00Problem Classes
Scheduled Learning And Teaching ActivitiesDrop-in/surgery101:0010:00Supervised Guided Learning
Guided Independent StudyIndependent study381:0038:00Background reading, coursework review, exam preparation
Total100:00
##### Jointly Taught With
Code Title
MAS3706Metric Spaces and Topology
##### Teaching Rationale And Relationship

The teaching methods are appropriate to allow students to develop a wide range of skills, from understanding basic concepts and facts to higher-order thinking. Lecture materials (video recordings, lecture notes) are used for the dissemination of theory and explanation of methods, illustrated with examples. Brief worksheets or online quizzes guide students’ independent study of the material and provide instant feedback on their understanding. Problem Classes provide the students with the opportunity to train the intended skill outcomes, receive feedback on their work, and ask questions. Drop-Ins provide students with the opportunity to fill the remaining gaps in their understanding and to receive guidance and hints to tackle the most difficult parts of the material.

#### Assessment Methods

The format of resits will be determined by the Board of Examiners

##### Exams
Description Length Semester When Set Percentage Comment
Written Examination1201A80N/A
##### Exam Pairings
Module Code Module Title Semester Comment
Topology1N/A
##### Other Assessment
Description Semester When Set Percentage Comment
Prob solv exercises1M5Problem-solving exercises assessment
Prob solv exercises1M5Problem-solving exercises assessment
Prob solv exercises1M5Problem-solving exercises assessment
Prob solv exercises1M5Problem-solving exercises assessment
##### Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. The format of the examination will enable students to reliably demonstrate their own knowledge, understanding and application of learning outcomes. The assurance of academic integrity forms a necessary part of programme accreditation.

Exam problems may require a synthesis of concepts and strategies from different sections, while they may have more than one ways for solution. The examination time allows the students to test different strategies, work out examples and gather evidence for deciding on an effective strategy, while carefully articulating their ideas and explicitly citing the theory they are using.

The coursework assignments allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; these assessments have a secondary formative purpose as well as their primary summative purpose.