ABSTRACT
This chapter introduces the reader to the notion of (arbitrary) metric space. The chapter starts with the set of real numbers, where the reader knows how to measure \distances" and gradually abstracts properties of this distance to define an abstract metric space. A lot of examples are provided and a variety of exercises are given in the text itself to clear the concepts easily. Throughout the text, three sets remain of interest to us: One, the set of real numbers and its Cartesian products (called the n-space), the set of sequences of certain type (see text for more information), and the set of continuous functions. Major remarks about these sets considered as metric spaces are given in this chapter. Immediately after defining metric spaces, we move towards the consequences of the definition. We look at open balls, open sets, closed sets, bounded sets, types of points in a set of a metric space and other related concepts. At the end of this chapter, we ask a major question concerning different metrics on the same set and try to answer it.
