Short notes on quasi-uniform spaces. II: Doubly uniformly strict extensions
type: Item
instance of: scholarly article
title: Short notes on quasi-uniform spaces. II: Doubly uniformly strict extensions
zbMATH Open document ID: 0841.54022
DOI: 10.1007/BF01874581
published in: Acta Mathematica Hungarica
publication date: 1996-07-18T00:00:00Z
review text: Before citing the main result let us mention the necessary terminology and facts. Throughout this review, \(X\) will denote a non-empty set, \({\mathcal U}\) a quasi-uniformity on \(X\) and \(X \subseteq Y\). A quasi-uniformity \({\mathcal V}\) on \(Y\) is an extension of \({\mathcal U}\) if \({\mathcal V} |X = {\mathcal U}\) and \(X\) is \({\mathcal T} ({\mathcal V})\)-dense in \(Y\); \({\mathcal V}\) is a double extension if, in addition, \(X\) is \({\mathcal T} ({\mathcal V}^{-1})\)-dense in \(Y\). For \(a \in Y\), \(f^1 (a)\) denotes the trace on \(X\) of the \({\mathcal T} ({\mathcal V})\)-neighborhood filter of \(a\), called the trace filter of \(a\); in the case of double extensions, \(f^{-1} (a)\) denotes the trace on \(X\) of the \({\mathcal T} ({\mathcal V}^{-1})\)-neighborhood filter of \(a\). Assume now that trace filters \(f^1(a)\) are prescribed, and one is looking for extensions of \({\mathcal U}\) inducing them. It is known that the following two conditions are necessary and sufficient for the existence of such an extension: (i) for \(x \in X\), \(f^1(x)\) is the \({\mathcal T} ({\mathcal U})\)-neighborhood filter of \(x\); (ii) each trace filter is round. Similarly, there is a double extension for prescribed filter pairs \((f^{-1} (a)\), \(f^1 (a))\) iff the following conditions hold: (i) for \(x \in X\), \((f^{-1} (x), f^1 (x))\) is the neighborhood filter pair of \(x\), (ii) each trace filter pair is round and Cauchy. In the following when trace filters (filter pairs, respectively) are prescribed it is always assumed that the above conditions are satisfied. The quasi-uniformity \({\mathcal V}\) on \(Y\) is called a uniformly strict extension of \({\mathcal U}\) provided that for each \(V \in {\mathcal V}\) there is a \(W \in {\mathcal V}\) such that \(s^1 (W(a) \cap X) \subseteq V(a)\) whenever \(a \in Y\), where \(s^i (A) = \{a \in Y : A \in f^i (a)\}\) \((A \subseteq X\), \(i \in \{-1, 1\})\). \({\mathcal V}\) is called a doubly uniformly strict…
Mathematics Subject Classification ID: 54E15
Mathematics Subject Classification ID: 54D35
zbMATH DE Number: 878109
zbMATH Keywords: doubly uniformly strict extension
zbMATH Keywords: doubly uniformly regular extension
author: J. Deak
reviewed by: Hans Peter Künzi
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 878109