On universal spaces and absorbing sets related to a transfinite extension of covering dimension

R. Pol has shown that for every countable ordinal number α there exists a universal space for separable metrizable spaces X with trind X ⩽ α . W. Olszewski has shown that for every countable limit ordinal number λ there is no universal space for separable metrizable space with trInd X ⩽ λ . T. Radul and M. Zarichnyi have proved that for every countable limit ordinal number there is no universal space for separable metrizable spaces with dim W X ⩽ α where dim W is a transfinite extension of covering dimension introduced by P. Borst. We prove the same result for another transfinite extension dim C of the covering dimension. application, we show that there is no absorbing sets (in the sense of Bestvina and Mogilski) for the classes of spaces X with dim C X ⩽ α belonging to some absolute Borel class.