p-adic numbers

Abstract

If X is a nonempty set, a distance, or metric, on X is a function d from pairs of elements (x, y) of X to the nonnegative real numbers such that

$$ \begin{gathered} d(x,\,y) = 0\,if\,and\,only\,if\,x = y. \hfill \\ \hfill \\ \end{gathered} $$

((1))

$$ d(x,\,y) = d(y,\,x). $$

((2))

$$ d(x,\,y)\, \leqslant \,d(x,\,z) + d(z,\,y)\,for\,all\,z\, \in \,X. $$

((3))

A set X together with a metrid d is called a metric space. The same set X can give rise to many different metric spaces (X, d), as we’ll soon see.