Module MA2223: Metric spaces
- Credit weighting (ECTS)
- 5 credits
- Semester/term taught
- Michaelmas term 2017-18
- 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 2-hour 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.