The new theory of ultrametric spaces (Q1922121)

The author considers ultrametric spaces \((X,d,\Gamma)\) where \((\Gamma, \leq)\) is a partially ordered set. Ultrametric spaces such that any (nonempty) chain of balls has a nonempty intersection are called spherically complete. A (unique) fixed point theorem for spherically complete ultrametric spaces is proved. To show the interest in this theorem some modern examples of these spaces are given. Moreover, a spherically complete space \(X^*\) is associated to every ultrametric space \(X\) in such a way that if \((\Gamma,\leq)\) is a totally ordered set, then there is a canonical embedding from \(X\) into \(X^*\).