Module Descriptors

Mathematics

MA341: Metric Spaces (5 ECTS)

This module introduces the theory of metric spaces. The real line with its natural notion of distance is a metric space, from which the metric space definition and theory readily evolves. Familiar concepts such as convergence and continuity are explored in this new broader context while new concepts and properties, such as closed sets and compactness, illuminate key basic facts about functions.

Taught in Semester(s) 1. Examined in Semester(s) 1.

Workload: 101 hours (24 Lecture hours, 12 Tutorial hours, 65 Self study hours).

Module Learning Outcomes. On successful completion of this module the learner should be able to:

  1. Write down, explain and use definitions of key concepts encountered throughout the module.
  2. Demonstrate how key definitions emerge naturally from the parent example given by the real line.
  3. Establish that each example from a given list forms a metric space and illustrate other properties which such examples may have.
  4. Construct proofs which connect and relate metric concepts.
  5. Produce examples which illustrate and distinguish definitions such as limit point of a set, complete metric space, closed set etc.
  6. Write down all mathematical work with rigour and precision.
  7. Create new or other lines of mathematical enquiry on the basis of mathematical ideas encountered in this module.

Indicative Content

This module introduces the theory of metric spaces with an emphasis on discovery learning by the student. Thus by developing familiarity and competence with the key building blocks (open balls, and then open sets) of the theory, students learn to forge connections and interrelations with ideas and concepts taught in previous years. The overall structure for the module is:

  1. Motivation, leading to
  2. Definition of a metric space; examples and non-examples. Make your own!
  3. New metric spaces (= subspaces) and new concepts (= continuous functions, convergent sequences) from old.
  4. Open sets, limit points, completeness, compactness.
  5. Application: Banach's Fixed point theorem (aka Contraction Mapping Theorem).
  6. Special subsets of the reals, including the Cantor set.

Module Resources

  1. Introduction to metric and topological spaces, by W. Sutherland, 2nd edition, Oxford University Press.
  2. Metric Spaces, by V. Bryant, Cambridge University Press.
  3. Introduction to Topology, by B. Mendelson, 3rd edition, Dover.
  4. Elementary topology, by M. Gemignani, 2nd edition, Dover.
  5. Metric Spaces, by M. Ó Searcóid, Springer SUMS.

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