Level of study Third/Final year
Credit value 20
Semester Students may study either 1 or both .
Pre-requisite modules 22497,22507
This module extends the ideas from sequences and series and differentiable functions from one real variable to several, and establishes and uses several main results in this area.
This module also 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 consider various ways in which a function can be the limit of a sequence of functions. Since the notion of topological space is, in itself, too abstract to be of much interest, extra structure is introduced: compactness, for instance, is shown to be the proper generalization of the closed bounded intervals so often used in the analysis of the real line. The course may end with an application: for example, using topological techniques one can prove the existence of continuous functions of the real line which fail to have derivatives at any point.