On successful completion of this module, students will be able to:

Accurately recall definitions, state theorems and produce proofs on topics in metric spaces normed vector spaces and topological spaces;

Construct rigorous mathematical arguments using appropriate concepts and terminology from the module, including open, closed and bounded sets, convergence, continuity, norm equivalence, operator norms, completeness, compactness and connectedness;

Solve problems by identifying and interpreting appropriate concepts and results from the module in specific examples involving metric, topological and /or normed vector spaces;

Construct examples and counterexamples related to concepts from the module which illustrate the validity of some prescribed combination of properties;

Module Content

Metric spaces (including open and closed sets, continuous maps and complete metric spaces);

Normed vector spaces (including operator norms and norms on finite dimensional vector spaces);

Topological properties of metric spaces (including Hausdorff, connected and compact spaces);

Module Prerequisite

Recommended Reading

Introduction to metric and topological spaces, W.A. Sutherland. Oxford University Press, 1975;

Metric Spaces, E.T. Copson. Cambridge University Press, 1968;

Assessment Detail

This module will be examined in a 2-hour examination in Trinity term. Continuous assessment will contribute 15% to the final grade for the module at the annual
examination session, with the examination counting for the remaining 85%. Supplemental exams if required will consist of 100% exam.