On immediate extensions of ultrametric spaces
type: Item
instance of: scholarly article
title: On immediate extensions of ultrametric spaces
zbMATH Open document ID: 0866.54028
DOI: 10.1007/BF03322231
author: Erwin Schörner
published in: Results in Mathematics
publication date: 1997-07-20T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 54E35
Mathematics Subject Classification ID: 12J20
zbMATH DE Number: 920818
zbMATH Keywords: ultrametric distance
zbMATH Keywords: pseudoconvergent sequence
zbMATH Keywords: pseudolimit
zbMATH Keywords: immediate extension
zbMATH Keywords: ultrametric space
zbMATH Keywords: valued fields
MaRDI profile type: Publication
cites work: Maximality and Ultracompleteness in Normed Modules
cites work: Maximal fields with valuations
cites work: Q3223453
cites work: The Banach fixed point theorem for ultrametric spaces
cites work: Fixed points, combs and generalized power series
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description: scientific article; zbMATH DE number 920818