Universal elements for some classes of spaces

Abstract

In the paper two dimensions, denoted by dm and Dm, are defined in the class of all Hausdorff spaces. The dimension Dm 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 elements. In we gave the dimension-like functions dmK,B E and DmK,B E, where K is a class of subsets, E a class of spaces and B a class of bases and we proved that in the families P(dm K, B E < K)and P(Dm K, B E < K) of all spaces X for which dm K,B E (X) < K and Dm K, B E (X) < K, 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 are universal elements.