Short notes on quasi-uniform spaces. II: Doubly uniformly strict extensions
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 extension of \({\mathcal U}\) provided that \({\mathcal V}\) is a uniformly strict extension of \({\mathcal U}\), and \({\mathcal V}^{-1}\) of \({\mathcal U}^{-1}\). A Cauchy filter pair \((f^{-1}, f^1)\) in \((X, {\mathcal U})\) is said to be weakly concentrated if for each \(U \in {\mathcal U}\) there is a \(V \in {\mathcal U}\) such that \(xUy\) whenever \(V(x) \in f^1\) and \(V^{-1} (y) \in f^{-1}\); a family of Cauchy filter pairs is uniformly weakly concentrated if the above condition holds for each filter pair, with \(V\) depending only on \(U\), but not on the filter pair. If trace filter pairs are prescribed then to each \(U \in {\mathcal U}\) an entourage \(^4U\) on \(Y\) is assigned as follows: \(a^4 Ub\) iff there are \(A \in f^{-1} (a)\), \(B \in f^1(b)\) with \(A \times B \subseteq U\). It is known that \(^4 {\mathcal U}\) is a quasi-uniformity iff the prescribed trace filter pairs are uniformly weakly concentrated. The main result of the paper is the following: There exists a doubly uniformly strict extension of a quasi-uniformity \({\mathcal U}\) for prescribed trace filter pairs iff they are uniformly weakly concentrated and minimal Cauchy; if so then \(^4 {\mathcal U}\) is the only doubly uniformly strict extension, and it is the coarsest double extension compatible with the trace filter pairs. The paper ends with related investigations on (doubly) uniformly regular extensions. It is known that each (doubly) uniformly regular extension is (doubly) uniformly strict. The following criterion is obtained. There is a doubly uniformly regular extension for prescribed trace filter pairs iff they are minimal Cauchy, and the (ultra)filter pairs finer than trace filter pairs are uniformly weakly concentrated. As a by-product of the investigations also the following two results are proved: A uniformly strict extension of a totally bounded quasi-uniformity is totally bounded. A quasi-uniformity is doubly uniformly regular iff the convergent filter pairs are uniformly weakly concentrated.
scientific article; zbMATH DE number 878109