Introduction

In previous courses you studied continuity of functions whose domain and range is the set of real numbers. Continuous functions where in this case defined in terms of the absolute value on the set of real numbers.
The absolute value is a distance function in the sense that if\( x\) and \(y\) are any two real numbers, then \(|x-y|\) gives the distance between \(x\) and \(y\).
In this course will introduce a general type of distance function, called a
metric, which will enable us to define continuity of functions in a more
abstract way.
This is desirable because we are then able to discuss continuity of functions defined on various types of sets not only sets of real numbers. We will go a step further and discuss continuity of functions in an even more general settings where the notation of distance is not used to define a continuous function. To achieve this, certain collections of sets called topologies, will play a prominent role.
In the light of previous paragraphs, this course is about metric and
topological spaces.
A metric space is a set \(X\) endowed with a metric \(d\) while a topological space is a set \(X\) together with a collection \(\tau \) of subsets of \(X\) that forms a topology.
Metric and topological spaces have applications in various branches of
mathematics and its applications.