School of Computing and Mathematics

Faculty of Natural Sciences

For academic year: 2019/20 Last Updated: 09 June 2019

MAT-30030 - Metric Spaces and Topology

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MAT-20029 Analysis II

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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.

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.

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.

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,

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,

Lectures: 24 hours

Examples Classes: 12 hours

Preparation of coursework: 24 hours

Independent study: 88 hours

Unseen examination: 2 hours

Examples Classes: 12 hours

Preparation of coursework: 24 hours

Independent study: 88 hours

Unseen examination: 2 hours

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Approximately eight assignments set at regular intervals.

Continuous assessment will consist of written coursework, problem sheets, class tests, or any combination thereof.

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.