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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 20% and the annual exam will count for 80%. Students who are required to take a supplemental exam will be assessed based on that exam alone.