Universal Borel mappings and Borel actions of groups
Let \(\mathbb D\) and \(\mathbb R\) be saturated classes of separable metrizable spaces. As examples of saturated classes, we point out the class of all separable metrizable spaces or the class of all separable metrizable spaces of locally finite dimension. Then the class of mappings of Borel class \(\alpha<\omega_1\) with domain in \(\mathbb D\) and range in \(\mathbb R\) contains a topologically universal mapping. Let \(G\) be a separable metrizable group. The class of all separable metrizable spaces \(X\in\mathbb D\) with actions of \(G\) of Borel class \(\alpha<\omega_1\) on them has a topologically universal element. Isometrical versions of the above results are also stated. All these results are improvements of former results of the author, where continuous mappings and actions were studied. The proofs use a construction of ``Containing Spaces'' from the author's book [Universal Spaces and Mappings, Noth-Holland Mathematics Studies, vol. 198, Elsevier (2005; Zbl 1072.54001)], where also the notion of saturated classes and several quoted results can be found.
scientific article; zbMATH DE number 5365216
scientific article