Metric Spaces

Abstract

This chapter is devoted to the study of a metric space in which a topology on a set X is generated by a non-negative real-valued scalar function called metric that may be interpreted as measuring some kind of a distance between any two elements, or points, of the set because some of its properties are quite reminiscent of the familiar notion of distance that we frequently encounter in daily life. This type of a topological space occupies a rather privileged position among all topological spaces because its topology is totally determined by a scalar distance function. We can safely presume that we are quite familiar with the properties of such a function and we are accustomed to deal effectively with it. Instead, a general topology is usually prescribed by some class of probably abstract subsets of an abstract set. The notion of a metric space was first introduced by Fréchet in 1906. However, the term metric space was coined by Hausdorff a little later.