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Topology

Overview

  • Credit value: 15 credits at Level 7
  • Assessment: a three-hour examination (80%) and short, problem-based coursework assignments (20%)

Module description

This module introduces you to the basic concepts of Point-Set Topology and Algebraic Topology including topological spaces and their invariants as well as the Fundamental Group and its higher dimensional analogues. In particular you will know by the end of the course how to determine if a topological space has basic topological invariants such as connectedness and compactness as well as how to compute fundamental groups. The course is stand alone but will also provide a grounding for those wanting to pursue research into many aspects of topology.

Indicative module syllabus

  • Definition and examples including the Euclidean, discrete and indiscrete topologies
  • Open, closed, clopen sets and neighbourhoods
  • Subspaces
  • Bases
  • Continuity, homeomorphisms and topological invariants
  • Path connected and connected spaces
  • Compact spaces and the Heine-Borel theorem
  • Products and quotients
  • Tychonov's theorem for finite products
  • Homotopy of continuous functions and homotopy equivalence between topological spaces
  • Simply connected spaces
  • The fundamental group of a space, homomorphisms induced by maps of spaces, change of base point, invariance under homotopy equivalence
  • Covering spaces and Deck transformations
  • Brief discussion of higher dimensional analogues such as homology and homotopy groups