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BSc Mathematics / Course details
Year of entry: 2023
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Course unit details:
|Unit level||Level 2|
|Teaching period(s)||Semester 2|
|Offered by||Department of Mathematics|
|Available as a free choice unit?||No|
A metric space is a set together with a good definition of the distance between each pair of points in the set. Metric spaces occur naturally in many parts of mathematics, including geometry, fractal geometry, topology, functional analysis and number theory. This lecture course will present the basic ideas of the theory, and illustrate them with a wealth of examples and applications.
This course unit is strongly recommended to all students who intend to study pure mathematics and is relevant to all course units involving advanced calculus or topology.
|Unit title||Unit code||Requirement type||Description|
|Sequences and Series||MATH10242||Pre-Requisite||Compulsory|
A good understanding of a foundational course, such as either version of Sets, Numbers and Functions is strongly recommended.
This course aims to develop basic ideas of a metric space (essentially a set with a reasonable idea of distance on it). We will study a variety of important examples, such as the space of continuous functions on [0,1] and the space of vertices of a graph. We will develop some key properties of metric spaces that allow us to extend some theorems from the real line to other settings. Finally we will prove the contraction mapping theorem, a very important theorem with applications across mathematics, physics and economics.
On completion of this unit successful students will be able to:
- Define several standard examples of metric spaces and prove simple results related to them.
- Determine whether a given metric space has any of the following properties: openness, closedness, completeness, compactness, path connectedness.
- Prove simple results related to all of the above notions, as well as that of continuity.
- State and prove the contraction mapping theorem along with some of its applications.
1.Basic Definitions. Euclidean metric, taxicab metric, discrete metric, edge metric, word metric, sup metric, L1 metric, Hausdorff metric, l2 metric, product metrics. Examples. [4 lectures]
2.Open and Closed Sets. Interior, closure, sequences and convergence, boundary. Denseness. Equivalent metrics. Examples. 
3.Uniform Convergence. Sequences of continuous functions. Examples. 
4.Continuous maps. Extending the elementary definition. Relationship with open sets, sequences. Examples 
5.Compactness. Open coverings. Continuous maps on compact sets. Compactness in Euclidean space. 
6.Completeness. Cauchy sequences. The Contraction Mapping Theorem, Examples. 
Coursework; Weighting within unit 20%
- End of semester examination; Weighting within unit 80%
Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding. Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback. Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.
Two books are particularly relevant. The first is
Wilson A. Sutherland, Introduction to Metric and Topological Spaces, Oxford University Press (Second Edition) 2009
which contains almost all the material in the course, is beautifully written, and is highly recommended. Copies are available to purchase in Blackwells, and to borrow from the JRUL. For an alternative view, try
Micheal O'Searcoid, Metric Spaces, Springer 2006.
|Scheduled activity hours|
|Independent study hours|
|Thomas Kempton||Unit coordinator|
The independent study hours will normally comprise the following. During each week of the taught part of the semester:
· You will normally have approximately 60-75 minutes of video content. Normally you would spend approximately 2-2.5 hrs per week studying this content independently
· You will normally have exercise or problem sheets, on which you might spend approximately 1.5hrs per week
· There may be other tasks assigned to you on Blackboard, for example short quizzes or short-answer formative exercises
· In some weeks you may be preparing coursework or revising for mid-semester tests
Together with the timetabled classes, you should be spending approximately 6 hours per week on this course unit.
The remaining independent study time comprises revision for and taking the end-of-semester assessment.