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.
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.
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)
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.
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
30 sessions of 1 hour (30%)
Seminars
9 sessions of 1 hour (9%)
Private study
61 hours (61%)
Total
100 hours
Private study description
reviewing lectured material and accompanying supplementary materials; working on both summative and formative coursework; revising for exams.
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 D1
Weighting
Study time
Eligible for self-certification
Problem sheets
15%
No
Centrally-timetabled examination (On-campus)
85%
No
2 hour examination - no books allowed
Answerbook Pink (12 page)
Assessment group R1
Weighting
Study time
Eligible for self-certification
In-person Examination - Resit
100%
No
Answerbook Pink (12 page)
Feedback on assessment
Marked homework (formative) is returned and discussed in smaller classes and exam feedback.
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.
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.
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)
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.
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
30 sessions of 1 hour (30%)
Seminars
9 sessions of 1 hour (9%)
Private study
61 hours (61%)
Total
100 hours
Private study description
reviewing lectured material and accompanying supplementary materials; working on both summative and formative coursework; revising for exams.
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 D1
Weighting
Study time
Eligible for self-certification
Problem sheets
15%
No
Centrally-timetabled examination (On-campus)
85%
No
2 hour examination - no books allowed
Answerbook Pink (12 page)
Assessment group R1
Weighting
Study time
Eligible for self-certification
In-person Examination - Resit
100%
No
Answerbook Pink (12 page)
Feedback on assessment
Marked homework (formative) is returned and discussed in smaller classes and exam feedback.
