MAT-30030 - Metric Spaces and Topology
Coordinator: Peter Fletcher Room: MAC2.36 Tel: +44 1782 7 33260
Lecture Time: See Timetable...
Level: Level 6
Credits: 15
Study Hours: 150
School Office: 01782 733075

Programme/Approved Electives for 2019/20

Mathematics Dual Honours (Level 6)


Available as a Free Standing Elective

No

Co-requisites

None

Prerequisites

MAT-20029 Analysis II


Barred Combinations

None


Description for 2019/20

This module builds on material from Level 5 Analysis to introduce the concepts of metric and topological spaces. A metric space is a set of mathematical objects of any kind, in which it is possible to define the `distance┐ between two objects. Concepts such as convergence of infinite sequences and continuous functions, which arose from the study of real numbers, can be generalised to metric spaces, making them more powerful and versatile. Topology is a further generalisation, in which there is no numerical measure of distance but simply a qualitative notion of `arbitrary closeness┐.
This module develops the following Keele Graduate attributes:
1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought.

4. The ability creatively to solve problems using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand.

6. The ability to communicate clearly and effectively in written and verbal forms.

Aims
The aim of this module is to study the abstraction of concepts in real analysis, through the study of metric and topological spaces, and to demonstrate why such abstraction is very fruitful.

Intended Learning Outcomes

determine whether or not a given structure is a metric space;: 1,2,

test sequences in a metric space for convergence and relate different notions of convergence on the same set;: 1,2,

state and apply the definitions of open, closed, interior and closure to given cases and prove their basic properties;: 1,2,

verify whether given functions are continuous. Relate different metric and topological characterisations of continuity;: 1,2,

verify that a given space is a topological space and prove properties of such spaces;: 1,2,

construct geometric examples (curves and surfaces) as quotient spaces and verify the homeomorphisms involved;: 1,2,

define a manifold and apply this to one dimensional and two dimensional examples;: 1,2,

construct and verify homeomorphisms in R^n; apply the concept to topological properties;: 1,2,

prove properties of completeness as needed for contraction mapping theorem.: 1,2,






















Study hours

Lectures: 24 hours
Examples Classes: 12 hours
Preparation of coursework: 24 hours
Independent study: 88 hours
Unseen examination: 2 hours


School Rules

None

Description of Module Assessment

1: Coursework weighted 30%
Approximately eight assignments set at regular intervals.
Continuous assessment will consist of written coursework, problem sheets, class tests, or any combination thereof.

2: Unseen Exam weighted 70%
Two-hour, end of module examination.
The examination paper will consist of no less than five and not more than eight questions, all of which are compulsory.