Uniform convergence of an infinite series of functions is then introduced and some important results established. The remainder of the module consists of an introduction to the language and concepts of metric spaces, a very important area of mathematics. A metric space is an abstract mathematical structure which is ideal for the study of convergence and continuity. Many problems in diverse areas of mathematics and its applications, when viewed through metric space spectacles, are seen to be fundamentally the same problem in disguise. This often leads to their solution.
Metric spaces are also of basic importance for the study of fractals and iterated function systems.
The purpose of this module is to give a general introduction to the theory of rings, which is a subject of central importance in algebra. Historically, some of the major discoveries have helped to shape the course of developments of modern abstract algebra. Today, ring theory is a possible meeting ground for many algebraic subdisciplines such as group theory, representation theory, Lie theory, algebraic geometry, homological algebra, to name but a few. It will also develop an axiomatic description of set theory to allow some discussion of the issues of completeness and consistency. This module extends and deepens the study of Complex Variables begun in Level 2 and starts with a recapitulation of the main results from that course that we shall need.