Topology and Advanced Analysis

Module code: G1026
Level 6
15 credits in autumn semester
Teaching method: Lecture
Assessment modes: Coursework, Unseen examination

This module will introduce you to some of the basic concepts and properties of topological spaces. The subject of topology has a central role in all of Mathematics and having a proper understanding of its concepts and main theorem is essential as part of an undergraduate mathematics curriculum.

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. 

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.