This module introduces metric and topological spaces as abstract settings for the study of analytical concepts such as convergence and continuity. This generalization allows one, for example, to regard functions as points of a space and to consider various ways in which the function can be the limit of other functions. Extra structure is introduced: compactness, for instance, is shown to be the proper generalization of the closed bounded intervals often used in analysis on the real line. Connectedness, completeness will also be introduced as will separation axioms, e.g. the Hausdorff property.
The course may end with some applications for example proving the existence of a function that is continuous everywhere on the real line, but differentiable nowhere. Also some methods from algebraic topology, e.g. homotopy, may be introduced.

Learning Outcomes

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

Understand the definitions of metric and topological spaces and verify these definitions in examples.

Understand convergence and continuity, and determine when sequences converge and when functions are continuous in examples.

Understand the definitions of compactness, connectedness, completeness and separation axioms, and prove basic results involving these properties.

Assessment

27722-01 : Raw Module Mark : Coursework (100%)

Assessment Methods & Exceptions

2 hour Written Unseen January Examination (80%); In-course Assessment (20%).