Analysis III – Metric and Topological Spaces
MODULE CODE
CREDIT VALUE
DELIVERY
Module Aims
Aim 1
The aim of this module is to introduce the students to concepts of point set topology by gradually generalizing familiar ideas from the analysis of the real line first to Euclidean spaces and then to more abstract metric and topological spaces.
Module Content
Metric and Normed Spaces: definitions and examples; balls and neighbourhoods; open and closed sets; limits and continuity; equivalent metric, Lipschitz equivalence; completeness; Banach’s fixed point theorem and applications.
Topological spaces: definitions and examples; open and closed sets; bases; continuity; homeomorphisms; subspaces; product spaces; connectedness; compactness; quotient spaces
Separation Axioms: Hausdorff spaces; regular and normal spaces; 1st and 2nd countable spaces
Learning Outcomes
On successful completion of this module, a student will be able to:
Teaching Methods
The module will be delivered on campus, with weekly lecture and tutorial sessions.
Printed notes will be given ahead of time for each section of the course, to support and enhance students’ preparation and engagement during class sessions. Lectures will follow the notes, with discussions of the main theoretical topics, and study of examples of the applications of the theory. There will be a strong emphasis on student involvement in discussions in lectures, to encourage a more active approach to learning the material, and to allow the delivery to be tailored to build on the students’ current understanding.
Regular formative work in tutorial sessions will allow students to internalise the mathematical ideas and methods developed in the lectures, and lead to the development of problem-solving skills. This formative work will also feed back into the delivery of lectures and tutorials.
Assessment Methods
The module is assessed through a Portfolio of exercises and an examination.
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Compulsory
Optional
