Ultra-quasi-metrically tight extensions of ultra-quasi-metric spaces
In [Adv. Math. 53, 321--402 (1984; Zbl 0562.54041)], \textit{A. W. M. Dress} has introduced the concept of tight extension of a metric space which is equivalent to a concept introduced by \textit{J. R. Isbell} [Comment. Math. Helv. 39, 65--76 (1964; Zbl 0151.30205)], namely the injective hull of a metric space. In 2014, the author together with \textit{P. Haihambo} and \textit{H.-P. A. Künzi} [in: V. Brattko (ed.) et al., Logic, computation, hierarchies. Dedicated to Victor L. Selivanov on the occasion of his 60th birthday. Berlin: De Gruyter. Ontos Mathematical Logic 4, 9--22 (2014; Zbl 1320.54018)] has constructed the tight extensions of quasi-metric spaces. In the present paper, the author does an appropriate modification, in order to come up with a tight extension for \(\mathbf{UQP}\)-metric spaces (\(T_0\)-ultra-quasi-metric spaces). Using techniques similar to those of Dress, the author shows that every \(\mathbf{UQP}\)-metric space \(X\) has a \(uq\)-tight extension which is maximal in the class of \(uq\)-tight extensions of \(X\).NEWLINENEWLINERemark that similar constructions in the category of \(T_0\)-ultra-quasi-metric spaces and contracting maps have been done by \textit{H.-P. A. Künzi} and \textit{O. O. Otafudu} [Appl. Categ. Struct. 21, No. 6, 651--670 (2013; Zbl 1284.54040)]. In this paper the author does the constructions in the category of \(\mathbf{UQP}\)-metric spaces and nonexpansive maps
scientific article; zbMATH DE number 6568846