Universal elements in some classes of mappings and classes of G-spaces

In the present paper it is proved the existence of topologically universal elements in some classes of mappings and classes of G-spaces generalizing some results from [S.D. Iliadis, Universal Spaces and Mappings, North-Holland Math. Stud., vol. 198, Elsevier, 2005] and [S.D. Iliadis, Universal Borel mappings and Borel actions of groups, Topology Appl., in press, doi:10.1016/j.topol.2007.05.029].