Universal elements in some classes of mappings and classes of \(G\)-spaces (Q958507)

The author proves a number of results on the existence of universal elements in classes of mappings between topological spaces. Given two mappings \(f\) and \(F\) between topological spaces and topological embeddings \(i: D_f \to D_F\) and \(j : R_f \to R_F\), say that the pair \((i,j)\) is a topological embedding of \(f\) into \(F\) if \(j \circ f = F \circ i\). Here \(D_g\) (\(R_g\), respectively) denotes the domain (the range, respectively) of the mapping \(g\). A mapping \(F\) is topologically universal in a class \({\mathbb F}\) of mappings between spaces if \(F \in {\mathbb F}\) and for every element of \({\mathbb F}\) there is a topological embedding into \(F\). A sample result runs as follows: given a saturated class of spaces \({\mathbb D}\), a \((\mu)\)-saturated class of spaces \({\mathbb R}\), and a \(\mu\)-additive complete saturated class of subsets \({\mathbb K}\), the class of all \({\mathbb K}\)-mappings with domain in \({\mathbb D}\) and range in \({\mathbb R}\) has topologically universal elements. A corollary is: given saturated classes of separable metrizable spaces \({\mathbb D}\) and \({\mathbb R}\), and an ordinal \(\alpha < \omega_1\), the class of all Borel mappings of additive class \(\alpha\) and with domain in \({\mathbb D}\) and range in \({\mathbb R}\) has topologically universal elements. Results on the existence of universal \(G\)-spaces where the topological group \(G\) acts on a space \(X\) are presented as well. This work generalizes earlier work of the author, see in particular his monograph [Universal spaces and mappings. North-Holland Mathematics Studies 198. Amsterdam: Elsevier (2005; Zbl 1072.54001)].