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Throughout the 2021-22 academic year, we will be prioritising face to face teaching as part of a blended learning approach that builds on the lessons learned over the course of the Coronavirus pandemic. Teaching will vary between online and on-campus delivery through the year, and you should read guidance from the academic department for details of how this will work for a particular module. You can find out more about the University’s overall response to Coronavirus at: https://warwick.ac.uk/coronavirus.

MA260-10 Norms, Metrics and Topologies

Department
Warwick Mathematics Institute
Level
Undergraduate Level 2
Module leader
David Wood
Credit value
10
Module duration
10 weeks
Assessment
Multiple
Study locations
  • University of Warwick main campus, Coventry Primary
  • Distance or Online Delivery
Introductory description

This is a module bridging Y1 Analysis and Y2 Analysis modules. The concepts of convergence, continuity, convergence and compactness are studied in the more general setting. This enables development of multi-dimensional and infinite-dimensional Analysis in consequent modules.

Module web page

Module aims

To introduce the notions of Normed Space, Metric Space and Topological Space, and the fundamental properties of Compactness, Connectedness and Completeness that they may possess.

Overall, this is an Analysis module, not a Topology module. The notion of topology is introduced but the focus is on the topologies, naturally occurring in Analysis. There will be no emphasis on topological spaces.

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

  • Recap of Y1 material: properties of intervals in R and basic geometry in R^n
  • Normed spaces: definitions, norms on R^n, spaces of linear operators, spaces of functions
  • Metric spaces: norms as metrics, metric on subsets, open and closed sets, convergence, continuity, uniform convergence of functions and applications (interchange of limits)
  • Topological spaces: basis, sub-basis, closure, interior, boundary, product topology, Hausdorff property, continuity, homeomorphisms and topological properties, topologically equivalent metrics
  • Connectedness: unions and products of connected sets, components, path-connected spaces, connected subsets of R and R^n
  • Compactness: Heine-Borel Theorem, equivalence of all norms on R^n, continuous functions on compact sets (Extreme Value Theorem and uniform continuity), sequential compactness of metric spaces
  • Completeness: R^n is complete, completion, Contraction Mapping Theorem, Arzela-Ascoli Theorem, applications to existence of solutions of ODEs
Learning outcomes

By the end of the module, students should be able to:

  • Demonstrate understanding of the basic concepts, theorems and calculations of Normed, Metric and Topological Spaces.
  • Demonstrate understanding of the open-set definition of continuity and its relation to previous notions of continuity, and applications to open or closed sets.
  • Demonstrate understanding of the basic concepts, theorems and calculations of the concepts of Compactness, Connectedness and Completeness (CCC).
  • Demonstrate understanding of the connections that arise between CCC, their relations under continuous maps, and simple applications.
Indicative reading list
  1. W A Sutherland, Introduction to Metric and Topological Spaces, OUP.
  2. E T Copson, Metric Spaces, CUP.
  3. G W Simmons, Introduction to Topology and Modern Analysis, McGraw Hill. (More advanced, although it starts at the beginning; helpful for several third year and MMath modules in analysis).
Subject specific skills

Familiarity with different ways of formulating convergence and continuity, and the relationship between them. Ability to use compactness and completeness arguments as part of larger proofs, frequently required in mathematical applications.

Transferable skills

Analytical and problem-solving skills as for any module in abstract mathematics. Facility for independent study and self motivation.

Study time

Type Required
Lectures 20 sessions of 1 hour (20%)
Seminars 5 sessions of 1 hour (5%)
Other activity 10 hours (10%)
Private study 65 hours (65%)
Total 100 hours
Private study description

self-working: reviewing lectured material and accompanying supplementary materials; working on both summative and formative coursework; revising for exams.

Other activity description

Collaborative project

Costs

No further costs have been identified for this module.

You do not need to pass all assessment components to pass the module.

Assessment group B
Weighting Study time
3 hour online examination (Summer) 100%

3 hour examination - no books allowed


  • Answerbook Pink (12 page)
Assessment group R
Weighting Study time
In-person Examination - Resit 100%
  • Answerbook Gold (24 page)
Feedback on assessment

Marked homework (formative) is returned and discussed in smaller classes and exam feedback.

Past exam papers for MA260

Courses

This module is Core for:

  • Year 2 of UMAA-G100 Undergraduate Mathematics (BSc)
  • Year 2 of UMAA-G103 Undergraduate Mathematics (MMath)

This module is Optional for:

  • Year 2 of UPXA-FG33 Undergraduate Mathematics and Physics (BSc MMathPhys)
  • Year 2 of UPXA-GF13 Undergraduate Mathematics and Physics (BSc)