Module MA2223: Metric spaces
 Credit weighting (ECTS)
 5 credits
 Semester/term taught
 Michaelmas term 201718
 Contact Hours
 11 weeks, 3 lectures including tutorials per week
 Lecturer
 Prof. Paschalis Karageorgis
 Learning Outcomes
 On successful completion of this module, students will be able to:
 accurately recall definitions, state theorems and produce proofs on topics in metric spaces, topological spaces and normed vector spaces;
 construct rigourous 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 spaces, topological spaces and normed vector spaces;
 construct examples and counterexamples related to concepts from the module which illustrate the validity of some prescribed properties.
 Module Content

The main concepts to be introduced in this module are the following.
 Metric spaces: metric, open ball, bounded, open set, convergence, closed set, continuity, Lipschitz continuity, pointwise and uniform convergence, Cauchy sequence, complete, contraction, completion, uniform continuity.
 Topological spaces: topology, metrisable, convergence, closed set, closure, interior, boundary, neighbourhood, limit point, continuity, subspace and product topology, Hausdorff, connected, compact, homeomorphism.
 Normed vector spaces: norm, bounded linear operator, operator norm, Euclidean norm, equivalent norms, Banach space, absolute convergence, invertible linear operator, dual space.
 Module Prerequisite
 MA1126 (Introduction to set theory and general topology).
 Recommended Reading

 Introduction to metric and topological spaces by Wilson Sutherland.
 Topology: a first course by James Munkres.
 Assessment
 This module will be examined in a 2hour examination in the Trinity term. Continuous assessment will count for 10% and the annual exam will count for 90%. Students who are required to take a supplemental exam will be assessed based on that exam alone.