Content
The first semester will cover general topology while the second semester will cover measure and integration. In more detail, the topics will be
- General Topology
- Introduction to general topology
- Limits
- Norms
- Metrics
- Open and closed balls
- Images and preimages
- Topologies
- Neighbourhoods
- Directed sets
- Filters and prefilters
- Other types of limit
- Sequences and nets
- Riemann integration
- Finite, countable and uncountable sets
- Cantor's Theorem
- The Schröder-Bernstein Theorem
- Zorn's Lemma
- Cardinality
- Countable and uncountable sets
- The Cantor set
- Disjoint unions and products
- Topological spaces
- Review
- Hausdorff spaces
- Interior, closure and boundary
- Dense subsets
- Continuity of functions
- Subspace topology
- Product topology
- Quotient topology
- Connectedness
- Compactness and σ-compactness
- The Heine-Borel Theorem
- Normal spaces
- Tietze Extension Theorem
- Urysohn's Lemma
- Partitions of unity
- Metric spaces
- Review
- Inequalities
- Examples
- Boundedness
- Lipschitz continuity
- Uniform convergence
- Local uniform convergence
- Cauchy sequences
- Completeness
- Completion
- The Banach Fixed Point Theorem
- Normed spaces
- Review
- Examples
- Bounded linear operators
- Operator norm
- Equivalence of norms
- Banach spaces
- Sums
- Dual space
- Filters and nets
- Review
- Nets
- Countable bases
- Measure and integration
- Introduction to measure and integration
- Jordan content and Riemann integration
- Measures and integrals
- The Lebesgue integral
- Convergence theorems
- Differentiation theorems
- The Fubini-Tonelli Theorem
The content will be a mix of general theory and specific examples.
Lectures
Lectures will be online due to Covid. Recordings and slides will be posted here and on Blackboard.
| Lecture | Date | Topic(s) | Video | Slides |
|---|---|---|---|---|
| 1 | 13 September 2021 | Introduction | Video | Slides |
| 2 | 13 September 2021 | Comments on reading and writing proofs | Video | Slides |
| 3 | 15 September 2021 | From R to metric spaces | Video | Slides |
| 4 | 20 September 2021 | Balls, images, preimages, open and closed sets | Video | Slides |
| 5 | 20 September 2021 | Topological spaces, Hausdorff topologies | Video | Slides |
| 6 | 23 September 2021 | Neighbourhoods | Video | Slides |
| 7 | 27 September 2021 | Where are we going? | Video | Slides |
| 8 | 27 September 2021 | Directed sets, monotone functions | Video | Slides |
| 9 | 30 September 2021 | Limits, sums and integrals | Video | Slides |
| 10 | 4 October 2021 | Introduction to Set Theory | Video | Slides |
| 11 | 4 October 2021 | Zorn's Lemma, Cardinality | Video | Slides |
| 12 | 7 October 2021 | Countability, the Cantor Set and Zorn again | Video | Slides |
| 13 | 11 October 2021 | Indexed collections of sets, products | Video | Slides |
| 14 | 11 October 2021 | Interior, closure boundary | Video | Slides |
| 15 | 14 October 2021 | Stronger and weaker topologies | Video | Slides |
| 16 | 18 October 2021 | Subspace and quotient topologies | Video | Slides |
| 17 | 18 October 2021 | Product topology | Video | Slides |
| 18 | 21 October 2021 | Connectedness | Video | Slides |
| 19 | 1 November 2021 | Compactness | Video | Slides |
| 20 | 1 November 2021 | More compactness | Video | Slides |
| 21 | 4 November 2021 | The Extreme Value Theorem, Bolzano-Weierstrass and normal spaces | Video | Slides |
| 22 | 8 November 2021 | Introduction to metric spaces | Video | Slides |
| 23 | 8 November 2021 | Boundedness, Lipschitz and uniform continuity | Video | Slides |
| 24 | 11 November 2021 | Boundedness, Lipschitz and uniform continuity II | Video | Slides |
| 25 | 15 November 2021 | Filters and convergence | Video | Slides |
| 26 | 15 November 2021 | Cauchy filters and completeness | Video | Slides |
| 27 | 18 November 2021 | Another compactness criterion, minimal Cauchy filters | Video | Slides |
| 28 | 22 November 2021 | Completion I | Video | Slides |
| 29 | 22 November 2021 | Completion II | Video | Slides |
| 30 | 25 November 2021 | The Banach Fixed Point Theorem | Video | Slides |
| 31 | 29 November 2021 | The Arzela-Ascoli Theorem | Video | Slides |
| 32 | 29 November 2021 | Normed vector spaces I | Video | Slides |
| 33 | 2 December 2021 | Normed vector spaces II | Video | Slides |
| 34 | 24 January 2022 | Comments on problem solving | Video | Slides |
| 35 | 25 January 2022 | Introduction to measure and integration | Video | Slides |
| 36 | 25 January 2022 | The extended real numbers | Video | Slides |
| 37 | 31 January 2022 | Convergence and absolute convergence of sums | Video | Slides |
| 38 | 1 February 2022 | Convergence theorems for limits of sums | Video | Slides |
| 39 | 1 February 2022 | More convergence theorems, sums of sums | Video | Slides |
| 40 | 7 February 2022 | Tonelli's and Fubini's theorems, Boolean algebras | Video | Slides |
| 41 | 8 February 2022 | σ-algebras, contents | Video | Slides |
| 42 | 8 February 2022 | Properties of contents | Video | Slides |
| 43 | 14 February 2022 | Jordan content, Banach-Tarski again, measures | Video | Slides |
| 44 | 15 February 2022 | Properties of measures | Video | Slides |
| 45 | 15 February 2022 | Completion of measures | Video | Slides |
| 46 | 21 February 2022 | Atomic algebras, partitions, equivalence relations | Video | Slides |
| 47 | 22 February 2022 | Morphisms of measure spaces, refinements | Video | Slides |
| 48 | 22 February 2022 | Definition of the integral, simple functions | Video | Slides |
| 49 | 28 February 2022 | Upper and lower integrals | Video | Slides |
| 50 | 1 March 2022 | Completions, Riemann integration, measurable functions | Video | Slides |
| 51 | 1 March 2022 | Convergence theorems for sequences of integrals | Video | Slides |
| 52 | 14 March 2022 | Riesz Representation Theorem, semicontinuous functions | Video | Slides |
| 53 | 15 March 2022 | Sketch of proof of Riesz Representation Theorem | Video | Slides |
| 54 | 15 March 2022 | Full version of Riesz Representation Theorem | Video | Slides |
| 55 | 21 March 2022 | Fundamental Theorem of Calculus (Overview) | Video | Slides |
| 56 | 22 March 2022 | First Fundamental Theorem of Calculus | Video | Slides |
| 57 | 24 March 2022 | Second Fundamental Theorem of Calculus (1/2) | Video | Slides |
| 58 | 28 March 2022 | Second Fundamental Theorem of Calculus (2/2) | Video | Slides |
| 59 | 29 March 2022 | Overview of construction of Lebesgue measure in higher dimensions | Video | Slides |
| 60 | 29 March 2022 | Convex geometry, convex polytopes and simplicial complexes | Video | Slides |
| 61 | 4 April 2022 | Jordan content in higher dimensions | Video | Slides |
| 62 | 6 April 2022 | Lebesgue measure in higher dimensions | Video | Slides |
| 63 | 6 April 2022 | Fubini's and Tonelli's Theorems | Video | Slides |
| 64 | 11 April 2022 | Module overview | Video | Slides |
| 65 | 12 April 2022 | The exam and how to revise for it | Video | Slides |
Text
There is no official textbook for the module but I will post notes here and on Blackboard.
Exams
There was a single exam, covering both terms, at the end of the second term. It will be worth 80% of your course mark. For further details on the structure and content of the exam see Lecture 65.
The past exams are available on the Exams Office website, but for some reason last year's exam is not. When looking at it keep in mind that it was an online open book exam, so somewhat untypical in various ways. I have also prepared a practice exam.Tutorials
There are weekly tutorials in the Joly Lecture Theatre on Fridays at 10:00.Assignments
There will be occasional assignments, which will count for 20% of your course mark. They will be posted here and on Blackboard. Solutions will be posted here once the assignment has been corrected.
| Assignment | Date Posted | Date Due | Problems | Solutions |
|---|---|---|---|---|
| 1 | 11 October 2021 | 20 October 2021 | Problems | Solutions |
| 2 | 15 November 2021 | 24 November 2021 | Problems | Solutions |
| 3 | 18 February 2022 | 1 March 2022 | Problems | Solutions |
| 4 | 4 April 2022 | 14 April 2022 | Problems | Solutions |