Mathematics

Topology and Advanced Analysis

Module code: 803G1
Level 7 (Masters)
15 credits in autumn teaching
Teaching method: Lecture
Assessment modes: Coursework, Unseen examination

Topics that will be covered in this module include:

  • Topological spaces
  • Base and sub-base
  • Separation axioms
  • Continuity
  • Metrisability
  • Completeness
  • Compactness and Coverings
  • Total Boundedness
  • Lebesgue numbers and Epsilon-nets
  • Sequential Compactness
  • Arzela-Ascoli Theorem
  • Montel's theorem
  • Infinite Products
  • Box and Product Topologies
  • Tychonov Theorem
  • Banach-Alaoglu theorem.

Module learning outcomes

  • Understand the concepts of a topological space, open and closed sets and interior and closure.
  • Understand the concept of a metric and a metrisable topology.
  • Understand and manipulate coverings, continuity and compactness.
  • Understand Arzela-Ascoli theorem and some of its basic consequences.