Metric Spaces and their Groups

Abstract

In many of the physical sciences a fundamental role is played by the concept of length: units of length are used to measure the distance between two points. In mathematics the idea of distance, as a function that assigns a real number to a given pair of points in some space, is formalised in terms of a few reasonablelooking properties, or axioms, and the result is called a metric on that space. Having defined a structure such as this on a set, it is natural to study those transformations, or maps, of such sets which preserve that structure. The requirement that these maps be invertible then leads naturally into the theory of groups.