MA26010 Norms, Metrics and Topologies
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 multidimensional and infinitedimensional Analysis in consequent modules.
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, subbasis, closure, interior, boundary, product topology, Hausdorff property, continuity, homeomorphisms and topological properties, topologically equivalent metrics
 Connectedness: unions and products of connected sets, components, pathconnected spaces, connected subsets of R and R^n
 Compactness: HeineBorel 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, ArzelaAscoli 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 openset 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
 W A Sutherland, Introduction to Metric and Topological Spaces, OUP.
 E T Copson, Metric Spaces, CUP.
 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 problemsolving 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
selfworking: 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

Assessment group R
Weighting  Study time  

Inperson Examination  Resit  100%  

Feedback on assessment
Marked homework (formative) is returned and discussed in smaller classes and exam feedback.
Courses
This module is Core for:
 Year 2 of UMAAG100 Undergraduate Mathematics (BSc)
 Year 2 of UMAAG103 Undergraduate Mathematics (MMath)
This module is Optional for:
 Year 2 of UPXAFG33 Undergraduate Mathematics and Physics (BSc MMathPhys)
 Year 2 of UPXAGF13 Undergraduate Mathematics and Physics (BSc)