Universal elements for some classes of spaces

In the paper [4] two dimensions, denoted by dm and Dm, are defined in the class of all fla,usdorff spaces. The dimension Din does not have the universality property in the class of separable metrizable spaces because the family of all such spaces X with Dm (X) <= 0 coincides with the family of all totally disconnected spaces in which there are no universal (see [5]). In [3] elements (see 1 we gave the dimension-like functions dm(E)(K,B) and Dm(E)(K,B), where IK is a class of subsets, lE a class of spaces and 1B a class of bases and we proved that in the families IP(dm(E)(K,B) < kappa) and P(Dm(E)(K,B) <= kappa) of all spaces X for which dm(E)(K,B)(X) < kappa and Dm(E)(K,B)(X) <= kappa, respectively there exist universal elements. In this paper, we give some new dimension-like functions and define using these definitions classes of spaces in which there arc universal elements.