Topology

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

In this module we introduce 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 module how to determine if a topological space has basic topological invariants such as connectedness and compactness as well as how to compute fundamental groups.

This module will provide a grounding in many aspects of topology if you want to pursue research.

Indicative 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