Pre-requisite: Students are expected to have studied material equivalent to that covered in the modules above.
OverviewThe module is intended to serve as an introduction to point-set topology, focusing on examples and applications. This will also enhance other modules by providing examples and concepts relevant to Functional Analysis, Algebra and Mathematical Physics.
The syllabus will include but is not restricted to topics from the following list:
• Basic definitions and examples (Euclidean and discrete spaces and non-metrizable examples such as the finite complement topology)
• Continuity and convergence in general topological spaces (especially related to the examples above)
• Product topology, subspace topology, quotient topology (including real and complex projective spaces)
• Compactness, including comparing different characterisations of compactness
• Homotopy and paths
• Homeomorphisms and homotopy equivalence, contractibility
• Connectedness and path-connectedness
• Winding number
• Fixed point theorems
In addition, for level 7 students:
• Advanced topic such as a topological proof of the Fundamental Theorem of Algebra; simply connected spaces.
Method of assessment
80% examination, 20% coursework.
The module will not follow a specific text. However, the following texts cover the material.
J.G. Hocking and G. Young: Topology, Dover Publications, 1988
J.R. Munkres: Topology, a first course, Prentice-Hall, 1975
C. Adams and A. Franzosa: Introduction to Topology, pure and applied, Pearson Prentice-Hall, 2008
The intended subject specific learning outcomes. On successfully completing the level 7 module students will be able to:
1 demonstrate systematic understanding of topology;
2 demonstrate the capability to solve complex problems using a very good level of skill in calculation and manipulation of the material in the following areas: topological spaces, continuity, convergence, homotopy theory, non-Euclidean geometry;
3 apply a range of concepts and principles in continuity and convergence in general topological spaces, path components and homotopy equivalence in loosely defined contexts, showing good judgment in the selection and application of tools and techniques.
The intended generic learning outcomes. On successfully completing the level 7 module students will be able to:
1 work competently and independently, be aware of their own strengths and understand when help is needed;
2 demonstrate a high level of capability in developing and evaluating logical arguments;
3 communicate arguments confidently with the effective and accurate conveyance of conclusions;
4 manage their time and use their organisational skills to plan and implement efficient and effective modes of working;
5 solve problems relating to qualitative and quantitative information;
6 make effective use of information technology skills such as online resources (Moodle), internet communication;
7 communicate technical material effectively;
8 demonstrate an increased level of skill in numeracy and computation;
9 demonstrate the acquisition of the study skills needed for continuing professional development.