Iterations of quasi-uniform hyperspace constructions
It is known, e.g. \textit{A. J. Ward} [Proc. Camb. Phil. Soc. 62, 207--208 (1966; Zbl 0137.16102)] and \textit{J. R. Isbell} [Proc. Camb. Phil. Soc. 62, 685--686 (1966; Zbl 0152.40001)], that two distinct uniformities \(\mathcal U\) and \(\mathcal V\) on a set \(X\) can give rise to Hausdorff uniformities \(\mathcal U_H\) and \(\mathcal V_H\) inducing the same topology on the set \(\mathcal P_0(X)\) of nonempty subsets of \(X\). On the other hand, \textit{V. Z. Polyakov} [Sov. Math. Dokl. 9, 318--319 (1968; Zbl 0172.48002)] has shown that if for two uniformities \(\mathcal U\) and \(\mathcal V\) on a set \(X\) the topologies induced by the Hausdorff uniformities \((\mathcal U_H)_H\) of \(\mathcal U_H\) and \((\mathcal V_H)_H\) of \(\mathcal V_H\) on \(\mathcal(\mathcal P_0(X))\) agree, then \(\mathcal U\) and \(\mathcal V\) must be equal. In the paper under review the author shows that the latter result does not generalize to quasi-uniformities, by providing two distinct quasi-uniformities \(\mathcal U\) and \(\mathcal V\) on the set \(\omega\) of nonnegative integers such that the iterated Hausdorff quasi-uniformities \((\mathcal U_H)_H\) and \((\mathcal V_H)_H\) induce the same topology on \(\mathcal(\mathcal P_0(X))\). Many interesting related questions are addressed throughout the article, some of which are stated as open problems.
scientific article; zbMATH DE number 5150235