Pseudo-uniformities
type: Item
instance of: scholarly article
title: Pseudo-uniformities
author: Tullio Valent
published in: Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Serie IX. Rendiconti Lincei. Matematica e Applicazioni
publication date: 2023-11-06T00:00:00Z
review text: In this paper a theory of pseudo-uniformity for a set \(X\) is considered and defined as a particular delta-local filter on \(X\times X\) such that its relations contain at least an element of the diagonal. Analogously to the classical theory of uniformity where the important result is the fact that each uniformity is generated by a family of pseudo-metrics, the author uses the same terminology for symmetric maps \(d : X\times X \mapsto \mathbb R^+\) satisfying the ``triangle inequality'' such that \(d(x,x)=0\) for some points \(x\in X\). Then an analogous theorem states that any delta-local filter on \(X\times X\) is defined by a family \(\mathcal P\) of pseudo-metrics with the property that for every \(x\in X\) there is \(d\in P\) such that \(d(x,x)=0\). Closely related to the former it is shown that any delta-local filter on \(X\times X\) induces a topology on \(X\). In Section 3 the author defines uniform continuity for maps with respect to delta-local filters and then the product of them is studied. In this context, the property of pseudo-uniform continuity for pseudo-metrics is considered. The main result of this treatment states that every pseudo-uniformity for \(X\) is defined by a family of pseudo-uniformly continuous pseudo-metrics on \(X\times X\). At the end it is pointed out that every topology \(\tau\) on \(X\) is defined by a family of pseudo-metrics \(d\) closely related to \(\tau\), and all topologies are pseudo-uniformizable.
reviewed by: Dieter Leseberg
Mathematics Subject Classification ID: 54E15
Mathematics Subject Classification ID: 54A05
Mathematics Subject Classification ID: 54E35
zbMATH DE Number: 7761232
zbMATH Keywords: general structure theory
zbMATH Keywords: uniform structures and generalizations
zbMATH Keywords: pseudo-uniformity
zbMATH Keywords: delta-local filter
zbMATH Keywords: pseudo-metric
zbMATH Keywords: pseudo-uniformizable
MaRDI profile type: Publication
cites work: Q3994935
cites work: Q3226265
DOI: 10.4171/RLM/998
description: scientific article; zbMATH DE number 7761232