On immediate extensions of ultrametric spaces
Let \(\Gamma\cup\{0\}\) be a totally ordered set with 0 as the first element, \(X\) be a nonempty set. A mapping \(d:X\times X\to \Gamma\cup \{0\}\) is called ultrametric distance if for all \(x,y,z\) from \(X\) the following axioms hold: (1) \(d(x,y)=0 \iff x=y\), (2) \(d(x,y)=d(y,x)\), (3) \(d(x,y)\leq \max(d(x,z), d(z,y))\). A pair \((X,d)\) is called an ultrametric space with value set \(\Gamma\). Let \(\rho\) denote a limit ordinal number. A sequence \((x_\delta)_{\delta<\rho}\) in \(X\) is called pseudoconvergent if \(d(x_\delta,x_{\delta'})> d(x_{\delta'},x_{\delta''})\) holds for all \(\delta< \delta'< \delta''<\rho\). For the pseudoconvergent sequence \((x_\delta)_{\delta<\rho}\) we get \(\pi_\delta: d(x_\delta,x_{\delta+1})= d(x_\delta,x_{\delta'})\) for all \(\delta<\delta'<\rho\); an element \(x\in X\) is called a pseudolimit of the sequence \((x_\delta)_{\delta<\rho}\) if \(d(x,x_\delta)= \pi_\delta\) holds for all \(\delta<\rho\). The ultrametric space \((X,d)\) is called pseudocomplete if every pseudoconvergent sequence of \((X,d)\) has a pseudolimit in \(X\). It is proved that an ultrametric space with value set is maximal if and only if it is pseudocomplete. Let \((Y,d)\) be an extension of an ultrametric space \((X,d)\). The extension \((Y,d)\) of \((X,d)\) is called an immediate extension if \(d(X\times X)= d(Y\times Y)\) and for all \(x\in X\), \(y\in Y\), \(x\neq y\), there exists an \(x'\in X\) such that \(d(x',y)< d(x,y)\). Every ultrametric space possesses a maximal immediate extension; it is unique within the class of so-called essential extensions. The most important examples of ultrametric spaces are given by valued fields.
scientific article; zbMATH DE number 920818
scientific article