Introductory description

The module treats various abstract settings in which to discuss convergence and continuity, and introduces concepts such as connectedness, compactness, and completeness, that will be taken further in future modules.

Norms provide a general measure of length, metrics a general measure of distance, and topologies an abstract setting in which to talk about "open sets". These notions are central in further work in both analysis and topology.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

Normed spaces - Metric spaces - Open and closed sets - Continuity - Topological spaces - The Hausdorff property and metrisability - Continuity between topological spaces - Compactness - Connectedness - Completeness in metric and normed spaces

Learning outcomes

By the end of the module, students should be able to:

• Demonstrate understanding of the basic concepts, theorems and calculations of Normed, Metric and Topological Spaces.
• Demonstrate understanding of the open-set definition of continuity and its relation to previous notions of continuity, and applications to open or closed sets.
• Demonstrate understanding of the basic concepts, theorems and calculations of the concepts of Compactness, Connectedness and Completeness (CCC).
• Demonstrate understanding of the connections that arise between CCC, their relations under continuous maps, and simple applications.

Indicative reading list

1. W A Sutherland, Introduction to Metric and Topological Spaces, OUP.
2. E T Copson, Metric Spaces, CUP.
3. G W Simmons, Introduction to Topology and Modern Analysis, McGraw Hill. (More advanced, although it starts at the beginning; helpful for several third year and MMath modules in analysis).

Subject specific skills

Familiarity with different ways of formulating convergence and continuity, and the relationship between them. Ability to use compactness and completeness arguments as part of larger proofs, frequently required in mathematical applications.

Transferable skills

Analytical and problem-solving skills as for any module in abstract mathematics. Facility for independent study and self motivation.