Urysohn universal space, its development and Hausdorff's approach
Three approaches to a direct construction of Urysohn universal space are compared, namely those of Urysohn, Hausdorff and Katětov. The focus is on Hausdorff's unpublished approach that is shown to work in a more general situation, and the following questions related to those constructions are added. {\parindent=6mm \begin{itemize}\item[1.] Katětov constructed a non-complete strongly \(\omega\)-universal and \(\omega\)-homogeneous space \(\mathcal{W}\). Can one describe such a space in terms of the space \(\mathcal{U}_\omega\)? \item[2.] Katětov proved that there is no \(\kappa\)-universal and \(\kappa\)-homogeneous space having density \(\kappa\) if \(\kappa <\kappa^{<\kappa}\). We believe that this result can be proved using \(\mathcal{U}_{\kappa}\). How do we do that? The author gives one possibility. \item[3.] V. Uspenskii used Katětov's construction for \(\kappa=\omega\) to prove universality of the group of isometries of Urysohn universal space. Is it possible to use the space \(\mathcal{U}_\kappa\) for similar results? \end{itemize}}
scientific article; zbMATH DE number 5306963