Introductory description

The concepts of convergence, continuity, convergence and compactness are studied in the more general setting. This enables development of multi-dimensional and infinite-dimensional Analysis in consequent modules.

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: definitions, norms on R^n, spaces of linear operators, spaces of functions
• Metric spaces: norms as metrics, metric on subsets, open and closed sets, convergence, continuity, uniform convergence of functions and applications (interchange of limits)
• Topological spaces: basis, sub-basis, closure, interior, boundary, product topology, Hausdorff property, continuity, homeomorphisms and topological properties, topologically equivalent metrics
• Connectedness: unions and products of connected sets, components, path-connected spaces, connected subsets of R and R^n
• Compactness: Heine-Borel Theorem, equivalence of all norms on R^n, continuous functions on compact sets (Extreme Value Theorem and uniform continuity), sequential compactness of metric spaces
• Completeness: R^n is complete, completion, Contraction Mapping Theorem, Arzela-Ascoli Theorem, applications to existence of solutions of ODEs

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. B. Mendelson (1991) Introduction to Topology. Dover.
4. 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.